JEE Mains Previous Year Questions (PYQ)

JEE Main 2022 27th July Morning Shift

An ellipse $ E:\; \displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $ passes through the vertices of the hyperbola $ H:\; \displaystyle\frac{x^{2}}{49}-\frac{y^{2}}{64}=-1 $. Let the major and minor axes of the ellipse $ E $ coincide with the transverse and conjugate axes of the hyperbola $ H $, respectively. Let the product of the eccentricities of $ E $ and $ H $ be $ {1}/{2} $. If $ l $ is the length of the latus rectum of the ellipse $ E $, then the value of $ 113l $ is equal to ______.

Let the sum of the focal distance of the point $ P(4,3) $ on the hyperbola $ H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 $ be $ 8\sqrt{\dfrac{5}{3}} $. If for $ H $, the length of the latus rectum is $ l $ and the product of the focal distances of the point $ P $ is $ m $, then $ 9l^2+6m $ is equal to:

7.

Let $ y=p(x) $ be the parabola passing through the points $ (-1,0), (0,1) $ and $ (1,0) $. If the area of the region $ \{(x,y):(x+1)^2+(y-1)^2\le 1,\; y\le p(x)\} $ is $ A $, then $ 12(\pi-4A) $ is equal to ________.

8.

If the ellipse $ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 $ meets the line $ \dfrac{x}{7}+\dfrac{y}{2\sqrt{6}}=1 $ on the x-axis and the line $ \dfrac{x}{7}-\dfrac{y}{2\sqrt{6}}=1 $ on the y-axis, then the eccentricity of the ellipse is:

9.

Let $ E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1 $ be an ellipse. Ellipses $ E_i $'s are constructed such that their centres and eccentricities are same as that of $ E_1 $, and the length of minor axis of $ E_i $ is the length of major axis of $ E_{i+1} $ $ (i \geq 1) $. If $ A_i $ is the area of the ellipse $ E_i $, then $ \frac{5}{\pi}\left( \displaystyle \sum_{i=1}^{\infty} A_i \right) $ is equal to ______.

10.

Let the focal chord of the parabola $P: y^{2}=4x$ along the line $L: y=mx+c,\, m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H: x^{2}-y^{2}=4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $OMFN$ is:

11.

The parabola $ y^2 = 4x $ divides the area of the circle $ x^2 + y^2 = 5 $ in two parts. The area of the smaller part is equal to:

12.

On the ellipse $ \frac{x^2}{8} + \frac{y^2}{4} = 1 $ let $ P $ be a point in the second quadrant such that the tangent at $ P $ to the ellipse is perpendicular to the line $ x + 2y = 0 $. Let $ S $ and $ S' $ be the foci of the ellipse and $ e $ be its eccentricity. If $ A $ is the area of the triangle $ SPS' $ then, the value of $ (5 - e^2)A $ is:

13.

Let $ PQ $ be a focal chord of the parabola $ y^2 = 4x $ such that it subtends an angle of $ \pi/2 $ at the point $ (3, 0) $. Let the line segment $ PQ $ be also a focal chord of the ellipse $ E : \, \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \, a^2 > b^2 $. If $ e $ is the eccentricity of the ellipse $ E $, then the value of $ \frac{1}{e^2} $ is equal to:

14.

Let $ P $ be a parabola with vertex $ (2,3) $ and directrix $ 2x+y=6 $. Let an ellipse $ E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1, a>b $, of eccentricity $ \dfrac{1}{\sqrt{2}} $ pass through the focus of the parabola $ P $. Then, the square of the length of the latus rectum of $ E $, is;

15.

Let a line $ L $ pass through the point of intersection of the lines $ bx+10y-8=0 $ and $ 2x-3y=0, \, b\in \mathbb{R}-\left\{ \dfrac{4}{3} \right\} $. If the line $ L $ also passes through the point $ (1,1) $ and touches the circle $ 17(x^{2}+y^{2})=16 $, then the eccentricity of the ellipse $ \dfrac{x^{2}}{5}+\dfrac{y^{2}}{b^{2}}=1 $ is:

16.

If the equation of the hyperbola with foci $ (4,2) $ and $ (8,2) $ is $ 3x^{2} - y^{2} - \alpha x + \beta y + \gamma = 0 $, then $ \alpha + \beta + \gamma $ is equal to __________.

17.

Consider a hyperbola $ H $ having center at the origin and foci on the $ x $-axis. Let $ C_{1} $ be the circle touching the hyperbola $ H $ and having the centre at the origin. Let $ C_{2} $ be the circle touching the hyperbola $ H $ at its vertex and having the centre at one of its foci. If areas (in sq unit) of $ C_{1} $ and $ C_{2} $ are $ 36\pi $ and $ 4\pi $, respectively, then the length (in units) of latus rectum of $ H $ is:

18.

Let $ E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,\, a > b $ and $ H : \frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1 $. Let the distance between the foci of $ E $ and the foci of $ H $ be $ 2\sqrt{3} $. If $ a - A = 2 $, and the ratio of the eccentricities of $ E $ and $ H $ is $ \frac{1}{3} $, then the sum of the lengths of their latus rectums is equal to:

19.

Let $ x = 2t,\, y = \frac{t^2}{3} $ be a conic. Let $ S $ be the focus and $ B $ be the point on the axis of the conic such that $ SA \perp BA $, where $ A $ is any point on the conic. If $ k $ is the ordinate of the centroid of $ \triangle SAB $, then $ \lim\limits_{t \to 1} k $ is equal to:

20.

If $ x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0 $ is the locus of a point, which moves such that it is always equidistant from the lines $ x + 2y + 7 = 0 $ and $ 2x - y + 8 = 0 $, then the value of $ g + c + h - f $ equals.

21.

Let the eccentricity of an ellipse $ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 $ is reciprocal to that of the hyperbola $ 2x^{2}-2y^{2}=1 $. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse ________ .

22.

The normal to the hyperbola $ \frac{x^2}{a^2}-\frac{y^2}{9}=1 $ at the point $ (8,\,3\sqrt{3}) $ on it passes through the point:

23.

Let $ P_{1} $ be parabola with vertex $ (3, 2) $ and focus $ (4, 4) $ and $ P_{2} $ be its mirror image with respect to the line $ x + 2y = 6 $. Then the directrix of $ P_{2} $ is $ x + 2y = $ ________ .

24.

Let $ T $ be the tangent to the ellipse $ E:\; x^{2}+4y^{2}=5 $ at the point $ P(1,1) $. If the area of the region bounded by the tangent $ T $, ellipse $ E $, lines $ x=1 $ and $ x=\sqrt{5} $ is $ \alpha \sqrt{5}+\beta+\gamma \cos ^{-1}\!\left(\dfrac{1}{\sqrt{5}}\right) $, then $ \left|\alpha+\beta+\gamma\right| $ is equal to _______.

25.

Let the product of the focal distances of the point $ (\sqrt{3}, \tfrac{1}{2}) $ on the ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, $ (a > b) $, be $ \tfrac{7}{4} $. Then the absolute difference of the eccentricities of two such ellipses is:

26.

Let the foci of a hyperbola coincide with the foci of the ellipse $ \frac{x^2}{36} + \frac{y^2}{16} = 1 $. If the eccentricity of the hyperbola is $ 5 $, then the length of its latus rectum is:

27.

Let $ H_{n}:\ \dfrac{x^{2}}{1+n}-\dfrac{y^{2}}{3+n}=1,\ n\in\mathbb{N} $. Let $ k $ be the smallest even value of $ n $ such that the eccentricity of $ H_{k} $ is a rational number. If $ l $ is the length of the latus rectum of $ H_{k} $, then $ 21l $ is equal to __________ .

28.

Let $ H $ be the hyperbola, whose foci are $ (1 \pm \sqrt{2}, 0) $ and eccentricity is $ \sqrt{2} $. Then the length of its latus rectum is:

29.

If $ x^{2} + 9y^{2} - 4x + 3 = 0, \, x, y \in \mathbb{R} $, then $ x $ and $ y $ respectively lie in the intervals:

30.

The distance of the point $ (6,-2\sqrt{2}) $ from the common tangent $ y=mx+c,\, m>0 $, of the curves $ x=2y^{2} $ and $ x=1+y^{2} $ is:

31.

Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^{2}=16x $ be $ (1,-4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m:n $, $ \gcd(m,n)=1 $, then $ m^{2}+n^{2} $ is equal to:

32.

If the chord joining the points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the parabola $y^2 = 12x$ subtends a right angle at the vertex of the parabola, then $x_1x_2 - y_1y_2$ is equal to:

33.

A line passing through the point $ P(\sqrt{5}, \sqrt{5}) $ intersects the ellipse $ \frac{x^2}{36} + \frac{y^2}{25} = 1 $ at $ A $ and $ B $ such that $ (PA)\cdot(PB) $ is maximum. Then $ 5(PA^2 + PB^2) $ is equal to:

34.

Let a line $ L_1 $ be tangent to the hyperbola $ \dfrac{x^2}{16} - \dfrac{y^2}{4} = 1 $ and let $ L_2 $ be the line passing through the origin and perpendicular to $ L_1 $. If the locus of the point of intersection of $ L_1 $ and $ L_2 $ is $ (x^2 + y^2)^2 = \alpha x^2 + \beta y^2 $, then $ \alpha + \beta $ is equal to _______.

35.

If the locus of the point, whose distances from the point $ (2, 1) $ and $ (1, 3) $ are in the ratio $ 5 : 4 $, is $ ax^{2} + by^{2} + cxy + dx + ey + 170 = 0 $, then the value of $ a^{2} + 2b + 3c + 4d + e $ is equal to:

36.

Let the focal chord $ PQ $ of the parabola $ y^{2} = 4x $ make an angle of $ 60^\circ $ with the positive $ x $-axis, where $ P $ lies in the first quadrant. If the circle, whose one diameter is $ PS $, $ S $ being the focus of the parabola, touches the $ y $-axis at the point $ (0, \alpha) $, then $ 5\alpha^{2} $ is equal to:

37.

Let the point $ P $ be the vertex of the parabola $ y = x^{2} - 6x + 12 $. If a line passing through the point $ P $ intersects the circle $ x^{2} + y^{2} - 2x - 4y + 3 = 0 $ at the point $ R $ and $ S $, then the maximum value of $ (PR + PS)^{2} $ is:

38.

Let $ H:\; \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 $ be a hyperbola such that the distance between its foci is $ 6 $ and the distance between its directrices is $ \frac{8}{3} $. If the line $ x=\alpha $ intersects the hyperbola H at the points $ A $ and $ B $ such that the area of the triangle $ AOB $ is $ 4\sqrt{15} $, where $ O $ is the origin, then $ \alpha^2 $ equals:

39.

A tangent line $ L $ is drawn at the point $ (2,-4) $ on the parabola $ y^{2}=8x $. If the line $ L $ is also tangent to the circle $ x^{2}+y^{2}=a $, then ‘$ a $’ is equal to ______.

40.

Let the normal at the point $ P $ on the parabola $ y^2 = 6x $ pass through the point $ (5, -8) $. If the tangent at $ P $ to the parabola intersects its directrix at the point $ Q $, then the ordinate of the point $ Q $ is:

41.

Let the latus rectum of the hyperbola $ \dfrac{x^2}{9} - \dfrac{y^2}{b^2} = 1 $ subtend an angle of $ \dfrac{\pi}{3} $ at the centre of the hyperbola. If $ b^2 $ is equal to $ \dfrac{l}{m}(1 + \sqrt{n}) $, where $ l $ and $ m $ are co-prime numbers, then $ l^2 + m^2 + n^2 $ is equal to $ \_\_\_\_\_\_\_\_ $.

42.

Let $ O $ be the origin, and $ P $ and $ Q $ be two points on the rectangular hyperbola $ xy = 12 $ such that the mid-point of the line segment $ PQ $ is $ \left( \frac{1}{2}, -\frac{1}{2} \right) $. Then the area of the triangle $ OPQ $ equals:

43.

Let $ C $ be the circle of minimum area enclosing the ellipse $ E:\; \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 $ with eccentricity $ \dfrac{1}{2} $ and foci $ (\pm 2,0) $. Let $ PQR $ be a variable triangle, whose vertex $ P $ is on the circle $ C $ and the side $ QR $ of length $ 2a $ is parallel to the major axis of $ E $ and contains the point of intersection of $ E $ with the negative $ y $-axis. Then the maximum area of the triangle $ PQR $ is:

44.

If the radius of the largest circle with centre $ (2,0) $ inscribed in the ellipse $ x^2 + 4y^2 = 36 $ is $ r $, then $ 12r^2 $ is equal to:

45.

Let the tangent drawn to the parabola $ y^2 = 24x $ at the point $ (\alpha, \beta) $ is perpendicular to the line $ 2x + 2y = 5 $. Then the normal to the hyperbola $ \dfrac{x^2}{\alpha^2} - \dfrac{y^2}{\beta^2} = 1 $ at the point $ (\alpha + 4, \beta + 4) $ does NOT pass through the point:

46.

Let $ A $ be the point $(3, 0)$ and circles with variable diameter $ AB $ touch the circle $ x^2 + y^2 = 36 $ internally. Let the curve $ C $ be the locus of the point $ B $. If the eccentricity of $ C $ is $ e $, then $ 72e^2 $ is equal to __________.

47.

Let $ ABCD $ be a trapezium whose vertices lie on the parabola $ y^2 = 4x $. Let the sides $ AD $ and $ BC $ of the trapezium be parallel to $ y $-axis. If the diagonal $ AC $ is of length $ \frac{25}{4} $ and it passes through the point $ (1, 0) $, then the area of $ ABCD $ is:

48.

If the $ x $-intercept of a focal chord of the parabola $ y^2 = 8x + 4y + 4 $ is $ 3 $, then the length of this chord is equal to $ \_\_\_\_ $.

49.

The equations of two sides of a variable triangle are $ x=0 $ and $ y=3 $, and its third side is a tangent to the parabola $ y^{2}=6x $. The locus of its circumcentre is:

50.

Let $ O $ be the vertex of the parabola $ x^2 = 4y $ and $ Q $ be any point on it. Let the locus of the point $ P $, which divides the line segment $ OQ $ internally in the ratio $ 2:3 $ be the conic $ C $. Then the equation of the chord of $ C $, which is bisected at the point $ (1, 2) $ is:

51.

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

52.

Let each of the two ellipses $ E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b) $ and $ E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, (A < B) $ have eccentricity $ \frac{4}{5} $. Let the lengths of the latus recta of $ E_1 $ and $ E_2 $ be $ l_1 $ and $ l_2 $, respectively, such that $ 2l_1^2 = 9l_2 $. If the distance between the foci of $ E_1 $ is $ 8 $, then the distance between the foci of $ E_2 $ is:

53.

Let the line $ L:\ \sqrt{2}x+y=\alpha $ pass through the point of the intersection $ P $ (in the first quadrant) of the circle $ x^2+y^2=3 $ and the parabola $ x^2=2y $. Let the line $ L $ touch two circles $ C_1 $ and $ C_2 $ of equal radius $ 2\sqrt{3} $. If the centres $ Q_1 $ and $ Q_2 $ of the circles $ C_1 $ and $ C_2 $ lie on the y-axis, then the square of the area of the triangle $ PQ_1Q_2 $ is equal to ______.

54.

If the eccentricity $ e $ of the hyperbola $ \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 $, passing through $ (6, 4\sqrt{3}) $, satisfies $ 15(e^{2} + 1) = 34e $, then the length of the latus rectum of the hyperbola $ \frac{x^{2}}{b^{2}} - \frac{y^{2}}{2(a^{2} + 1)} = 1 $ is:

55.

Let a line $ L: 2x + y = k,\ k > 0 $ be a tangent to the hyperbola $ x^2 - y^2 = 3 $. If $ L $ is also a tangent to the parabola $ y^2 = \alpha x $, then $ \alpha $ is equal to:

56.

Let the foci of a hyperbola be $ (1, 14) $ and $ (1, -12) $. If it passes through the point $ (1, 6) $, then the length of its latus-rectum is:

57.

For real numbers $ a, b $ ($ a > b > 0 $), let
$ \text{Area}\left\{(x,y): x^2 + y^2 \le a^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \ge 1 \right\} = 30\pi $ and $ \text{Area}\left\{(x,y): x^2 + y^2 \ge b^2 \text{ and } \frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1 \right\} = 18\pi $.
Then the value of $ (a-b)^2 $ is equal to ______.

58.

Let $ A(0,1), B(1,1) $ and $ C(1,0) $ be the mid-points of the sides of a triangle with incentre at the point $ D $. If the focus of the parabola $ y^{2}=4ax $ passing through $ D $ is $ (\alpha+\beta\sqrt{2},0) $, where $ \alpha $ and $ \beta $ are rational numbers, then $ \dfrac{\alpha}{\beta^{2}} $ is equal to:

59.

A common tangent $ T $ to the curves $ C_{1}:\ \dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1 $ and $ C_{2}:\ \dfrac{x^{2}}{42}-\dfrac{y^{2}}{143}=1 $ does not pass through the fourth quadrant. If $ T $ touches $ C_{1} $ at $ (x_{1},y_{1}) $ and $ C_{2} $ at $ (x_{2},y_{2}) $, then $ |2x_{1}+x_{2}| $ is equal to ______.

60.

Let $ P\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), Q, R $ and $ S $ be four points on the ellipse $ 9x^2 + 4y^2 = 36 $. Let $ PQ $ and $ RS $ be mutually perpendicular and pass through the origin. If $ \dfrac{1}{(PQ)^2} + \dfrac{1}{(RS)^2} = \dfrac{p}{q} $, where $ p $ and $ q $ are coprime, then $ p+q $ is equal to:

61.

If the length of the latus rectum of a parabola, whose focus is $ (a,a) $ and the tangent at its vertex is $ x+y=a $, is $ 16 $, then $ |a| $ is equal to:

62.

Let $ E $ be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at $ (3,-4) $, one focus at $ (4,-4) $ and one vertex at $ (5,-4) $. If $ mx-y=4, \ m>0 $ is a tangent to the ellipse $ E $, then the value of $ 5m^{2} $ is equal to ____________.

63.

Consider the parabola with vertex $ \left( \frac{1}{2}, \frac{3}{4} \right) $ and the directrix $ y = \frac{1}{2} $. Let $ P $ be the point where the parabola meets the line $ x = -\frac{1}{2} $. If the normal to the parabola at $ P $ intersects the parabola again in the point $ Q $, then $ (PQ)^2 $ is equal to:

64.

For some $ \theta \in \left(0, \frac{\pi}{2}\right) $, let the eccentricity and the length of the latus rectum of the hyperbola $ x^2 - y^2 \sec^2 \theta = 8 $ be $ e_1 $ and $ l_1 $, respectively, and let eccentricity and the length of the latus rectum of the ellipse $ x^2 \sec^2 \theta + y^2 = 6 $ be $ e_2 $ and $ l_2 $, respectively. If $ e_1^2 = e_2^2 \left(\sec^2 \theta + 1\right) $, then $ \left|\dfrac{l_1 l_2}{e_1 e_2}\right| \tan^2 \theta $ is equal to __________.

65.

Let the tangent to the parabola $ y^{2}=12x $ at the point $ (3,\alpha) $ be perpendicular to the line $ 2x+2y=3 $. Then the square of distance of the point $ (6,-4) $ from the normal to the hyperbola $ \alpha^{2}x^{2}-9y^{2}=9\alpha^{2} $ at its point $ (\alpha-1,\alpha+2) $ is equal to ______.

66.

For $ 0<\theta<\frac{\pi}{2} $, if the eccentricity of the hyperbola $ x^2-y^2cosec^2\theta=5 $ is $ \sqrt{7} $ times eccentricity of the ellipse $ x^2cosec^2\theta+y^2=5 $, then the value of $ \theta $ is:

67.

Let the line $ 2x+3y-k=0,\,k>0 $, intersect the $ x $-axis and $ y $-axis at the points $ A $ and $ B $, respectively. If the equation of the circle having the line segment $ AB $ as a diameter is $ x^{2}+y^{2}-3x-2y=0 $ and the length of the latus rectum of the ellipse $ x^{2}+9y^{2}=k^{2} $ is $ \frac{m}{n} $, where $ m $ and $ n $ are coprime, then $ 2m+n $ is equal to:

68.

If the line $ 3x - 2y + 12 = 0 $ intersects the parabola $ 4y = 3x^2 $ at the points $ A $ and $ B $, then at the vertex of the parabola, the line segment $ AB $ subtends an angle equal to:

69.

Let the length of the focal chord $ PQ $ of the parabola $ y^2 = 12x $ be $ 15 $ units. If the distance of $ PQ $ from the origin is $ p $, then $ 10p^2 $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

70.

The line $ x = 8 $ is the directrix of the ellipse $ E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 $ with the corresponding focus $ (2, 0) $. If the tangent to $ E $ at the point $ P $ in the first quadrant passes through the point $ (0, 4\sqrt{3}) $ and intersects the $ x $-axis at $ Q $, then $ (3PQ)^{2} $ is equal to ______.

71.

Let the tangent and normal at the point $ (3\sqrt{3}, 1) $ on the ellipse $ \dfrac{x^2}{36} + \dfrac{y^2}{4} = 1 $ meet the y-axis at the points $ A $ and $ B $ respectively. Let the circle $ C $ be drawn taking $ AB $ as a diameter and the line $ x = 2\sqrt{5} $ intersect $ C $ at the points $ P $ and $ Q $. If the tangents at the points $ P $ and $ Q $ on the circle intersect at the point $ (\alpha, \beta) $, then $ \alpha^2 - \beta^2 $ is equal to:

72.

Let the image of parabola $ x^{2}=4y $ in the line $ x-y=1 $ be $ (y+a)^{2}=b(x-c) $, $ a,b,c \in \mathbb{N} $. Then $ a+b+c $ is equal to:

73.

If two tangents drawn from a point $ P $ to the parabola $ y^{2}=16(x-3) $ are at right angles, then the locus of point $ P $ is:

74.

The vertices of a hyperbola $ H $ are $ (\pm 6, 0) $ and its eccentricity is $ \frac{\sqrt{5}}{2} $. Let $ N $ be the normal to $ H $ at a point in the first quadrant and parallel to the line $ \sqrt{2}x + y = 2\sqrt{2} $. If $ d $ is the length of the line segment of $ N $ between $ H $ and the $ y $-axis then $ d^2 $ is equal to ______ .

75.

Let the ellipse $ E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\, a > b $ and $ E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1,\, A < B $ have same eccentricity $ \frac{1}{\sqrt{3}} $. Let the product of their lengths of latus rectums be $ \frac{32}{\sqrt{3}} $, and the distance between the foci of $ E_1 $ be $ 4 $. If $ E_1 $ and $ E_2 $ meet at $ A $, $ B $, $ C $ and $ D $, then the area of the quadrilateral $ ABCD $ equals:

76.

Let the ellipse $3x^2 + py^2 = 4$ pass through the centre $C$ of the circle $x^2 + y^2 - 2x - 4y - 11 = 0$ of radius $r$. Let $f_1, f_2$ be the focal distances of the point $C$ on the ellipse. Then $6 f_1 f_2 - r$ is equal to:

77.

Let $ O $ be the vertex of the parabola $ y^{2}=4x $ and its chords $ OP $ and $ OQ $ are perpendicular to each other. If the locus of the mid-point of the line segment $ PQ $ is a conic $ C $, then the length of its latus rectum is:

78.

In a group of $ 100 $ persons $ 75 $ speak English and $ 40 $ speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $ \alpha $ and the number of persons who speak only Hindi is $ \beta $, then the eccentricity of the ellipse $ 25(\beta^{2}x^{2}+a^{2}y^{2})=\alpha^{2}\beta^{2} $ is:

79.

If $ S $ and $ S' $ are the foci of the ellipse $ \frac{x^{2}}{18} + \frac{y^{2}}{9} = 1 $ and $ P $ be a point on the ellipse, then $ \min(SP \cdot S'P) + \max(SP \cdot S'P) $ is equal to:

80.

Let the shortest distance from $ (a, 0), a > 0 $, to the parabola $ y^{2} = 4x $ be $ 4 $. Then the equation of the circle passing through the point $ (a, 0) $ and the focus of the parabola, the having its centre on the axis of the parabola is:

81.

Let one focus of the hyperbola $ H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $ and the corresponding directrix be $ x = \frac{9}{\sqrt{10}} $. If $ e $ and $ l $ respectively are the eccentricity and the length of the latus rectum of $ H $, then $ 9(e^2 + l) $ is equal to:

82.

Let the eccentricity of the hyperbola $ H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be $ \sqrt{\frac{5}{2}} $ and length of its latus rectum be $ 6\sqrt{2} $. If $ y = 2x + c $ is a tangent to the hyperbola $ H $, then the value of $ c^2 $ is equal to:

83.

Let $ A (\sec \theta, 2 \tan \theta) $ and $ B (\sec \phi, 2 \tan \phi) $, where $ \theta + \phi = \frac{\pi}{2} $, be two points on the hyperbola $ 2x^{2} - y^{2} = 2 $. If $ (\alpha, \beta) $ is the point of the intersection of the normals to the hyperbola at $ A $ and $ B $, then $ (2\beta)^{2} $ is equal to ________.

84.

Let $ L_1, L_2 $ be the lines passing through the point $ P(0,1) $ and touching the parabola $ 9x^2 + 12x + 18y - 14 = 0 $. Let $ Q $ and $ R $ be the points on the lines $ L_1 $ and $ L_2 $ such that the $ \triangle PQR $ is an isosceles triangle with base $ QR $. If the slopes of the lines $ QR $ are $ m_1 $ and $ m_2 $, then $ 16\left(m_1^2 + m_2^2\right) $ is equal to ____.

85.

Let the common tangents to the curves $ 4(x^{2}+y^{2}) = 9 $ and $ y^{2} = 4x $ intersect at the point $ Q $. Let an ellipse, centered at the origin $ O $, has lengths of semi-minor and semi-major axes equal to $ OQ $ and $ 6 $, respectively. If $ e $ and $ l $ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $ \frac{l}{e^{2}} $ is equal to______________.

86.

Let $ PQ $ be a chord of the hyperbola $ \frac{x^2}{4}-\frac{y^2}{b^2}=1 $, perpendicular to the $ x $-axis such that $ OPQ $ is an equilateral triangle, $ O $ being the centre of the hyperbola. If the eccentricity of the hyperbola is $ \sqrt{3} $, then the area of the triangle $ OPQ $ is:

87.

Let $ PQ $ be a chord of the parabola $ y^2 = 12x $ and the midpoint of $ PQ $ be at $ (4, 1) $. Then, which of the following point lies on the line passing through the points $ P $ and $ Q $?

88.

If two tangents drawn from a point $ (\alpha, \beta) $ lying on the ellipse $ 25x^{2} + 4y^{2} = 1 $ to the parabola $ y^{2} = 4x $ are such that the slope of one tangent is four times the other, then the value of $ (10\alpha + 5)^{2} + (16\beta^{2} + 50)^{2} $ equals ________________.

89.

If the tangent at a point $ P $ on the parabola $ y^{2}=3x $ is parallel to the line $ x+2y=1 $ and the tangents at the points $ Q $ and $ R $ on the ellipse $ \dfrac{x^{2}}{4}+\dfrac{y^{2}}{1}=1 $ are perpendicular to the line $ x-y=2 $, then the area of the triangle $ PQR $ is:

90.

Let $ T $ and $ C $ respectively be the transverse and conjugate axes of the hyperbola $ 16x^{2}-y^{2}+64x+4y+44=0 $. Then the area of the region above the parabola $ x^{2}=y+4 $, below the transverse axis $ T $ and on the right of the conjugate axis $ C $ is:

91.

Let the tangents at the points $ P $ and $ Q $ on the ellipse $ \dfrac{x^{2}}{2} + \dfrac{y^{2}}{4} = 1 $ meet at the point $ R(\sqrt{2},\, 2\sqrt{2} - 2) $. If $ S $ is the focus of the ellipse on its negative major axis, then $ SP^{2} + SQ^{2} $ is equal to __________________.

92.

Consider the circle $ C: x^2 + y^2 = 4 $ and the parabola $ P: y^2 = 8x $. If the set of all values of $ \alpha $, for which three chords of the circle $ C $ on three distinct lines passing through the point $ (\alpha, 0) $ are bisected by the parabola $ P $ is the interval $ (p, q) $, then $ (2q - p)^2 $ is equal to ________.

93.

Let $ R $ be the focus of the parabola $ y^2 = 20x $ and the line $ y = mx + c $ intersect the parabola at two points $ P $ and $ Q $. Let the point $ G(10, 10) $ be the centroid of the triangle $ PQR $. If $ c - m = 6 $, then $ (PQ)^2 $ is:

94.

Let $ e_{1} $ and $ e_{2} $ be two distinct roots of the equation $ x^{2}-ax+2=0 $. Let the sets $ \{\, a \in \mathbb{R} : e_{1} $ and $ e_{2} $ are the eccentricities of hyperbolas \} $ = (\alpha,\beta) $, and $ \{\, a \in \mathbb{R} : e_{1} $ and $ e_{2} $ are the eccentricities of an ellipse and a hyperbola, respectively \} $ = (\gamma,\infty) $.
Then $ \alpha^{2}+\beta^{2}+\gamma^{2} $ is equal to:

95.

An ellipse $ E:\; \displaystyle\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $ passes through the vertices of the hyperbola $ H:\; \displaystyle\frac{x^{2}}{49}-\frac{y^{2}}{64}=-1 $. Let the major and minor axes of the ellipse $ E $ coincide with the transverse and conjugate axes of the hyperbola $ H $, respectively. Let the product of the eccentricities of $ E $ and $ H $ be $ {1}/{2} $. If $ l $ is the length of the latus rectum of the ellipse $ E $, then the value of $ 113l $ is equal to ______.

96.

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $ x $-axis at a distance $ R $ and $ S $ ($ > R $) respectively from the origin, is:

97.

The foci of a hyperbola are $ (\pm 2, 0) $ and its eccentricity is $ \dfrac{3}{2} $. A tangent, perpendicular to the line $ 2x + 3y = 6 $, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x and y-axes are $ a $ and $ b $ respectively, then $ |6a| + |5b| $ is equal to _________.

98.

The length of the latus rectum and directrices of a hyperbola with eccentricity $ e $ are $ 9 $ and $ x = \pm \frac{4}{\sqrt{3}} $, respectively. Let the line $ y-\sqrt{3}x+\sqrt{3}=0 $ touch this hyperbola at $ (x_0,y_0) $. If $ m $ is the product of the focal distances of the point $ (x_0,y_0) $, then $ 4e^2+m $ is equal to ______.

99.

Let $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $, $ a>b $ be an ellipse, whose eccentricity is $ \frac{1}{\sqrt{2}} $ and the length of the latusrectum is $ \sqrt{14} $. Then the square of the eccentricity of $ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $ is:

100.

The focus of the parabola $ y^2 = 4x + 16 $ is the centre of the circle $ C $ of radius $ 5 $. If the value of $ \lambda $, for which $ C $ passes through the point of intersection of the lines $ 3x - y = 0 $ and $ x + \lambda y = 4 $, are $ \lambda_1 $ and $ \lambda_2 $, $ \lambda_1 < \lambda_2 $, then $ 12\lambda_1 + 29\lambda_2 $ is equal to __________.

101.

The line $ 12x \cos \theta + 5y \sin \theta = 60 $ is tangent to which of the following curves?

102.

Let $ P(a,b) $ be a point on the parabola $ y^{2}=8x $ such that the tangent at $ P $ passes through the centre of the circle $ x^{2}+y^{2}-10x-14y+65=0 $. Let $ A $ be the product of all possible values of $ a $ and $ B $ be the product of all possible values of $ b $. Then the value of $ A+B $ is equal to:

103.

Let a circle of radius $ 4 $ be concentric to the ellipse $ 15x^2+19y^2=285 $. Then the common tangents are inclined to the minor axis of the ellipse at the angle.

104.

If the points of intersection of the ellipse $ \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 $ and the circle $ x^2 + y^2 = 4b,\, b>4 $ lie on the curve $ y^2 = 3x^2 $, then $ b $ is equal to:

105.

The length of the chord of the ellipse $ \dfrac{x^2}{25} + \dfrac{y^2}{16} = 1 $, whose mid-point is $ \left( 1, \dfrac{2}{5} \right) $, is equal to:

106.

Let $ A $, $ B $ and $ C $ be three points on the parabola $ y^2 = 6x $ and let the line segment $ AB $ meet the line $ L $ through $ C $ parallel to the $ x $-axis at the point $ D $. Let $ M $ and $ N $ respectively be the feet of perpendiculars from $ A $ and $ B $ on $ L $. Then $ \left(\frac{AM \cdot BN}{CD}\right)^2 $ is equal to ________.

107.

A circle with centre $ (2, 3) $ and radius $ 4 $ intersects the line $ x + y = 3 $ at the points $ P $ and $ Q $. If the tangents at $ P $ and $ Q $ intersect at the point $ S(\alpha, \beta) $, then $ 4\alpha - 7\beta $ is equal to _________.

108.

Let a tangent to the curve $ y^{2}=24x $ meet the curve $ xy=2 $ at the points $ A $ and $ B $. Then the mid-points of such line segment $ AB $ lie on a parabola with the:

109.

Let the directrix of the parabola $ P: y^{2} = 8x $, cut x-axis at the point $ A $. Let $ B(\alpha, \beta),\ \alpha > 1 $, be a point on $ P $ such that the slope of $ AB $ is $ \frac{3}{5} $. If $ BC $ is a focal chord of $ P $, then six times the area of $ \triangle ABC $ is:

110.

If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y=4x-1$, then the co-ordinates of $P$ are:

111.

Let $ E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, \ a > b. $ Let $ E_2 $ be another ellipse such that it touches the end points of major axis of $ E_1 $ and the foci of $ E_2 $ are the end points of minor axis of $ E_1 $. If $ E_1 $ and $ E_2 $ have same eccentricities, then its value is:

112.

Let $ A $, $ B $ and $ C $ be the vertices of a variable right angled triangle inscribed in the parabola $ y^2 = 16x $. Let the vertex $ B $ containing the right angle be $ (4, 8) $ and the locus of the centroid of $ \Delta ABC $ be a conic $ C_0 $. Then three times the length of latus rectum of $ C_0 $ is ________.

113.

The locus of the centroid of the triangle formed by any point $ P $ on the hyperbola $ 16x^{2}-9y^{2}+32x+36y-164=0 $ and its foci is:

114.

If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$ and eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2}, \frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to:

115.

The tangents at the points $ A(1, 3) $ and $ B(1, -1) $ on the parabola $ y^2 - 2x - 2y = 1 $ meet at the point $ P $. Then the area (in $ \text{unit}^2 $) of the triangle $ PAB $ is:

116.

Let the length of the latus rectum of an ellipse $ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ (a>b), $ be $ 30 $. If its eccentricity is the maximum value of the function $ f(t)=-\dfrac{3}{4}+2t-t^{2} $, then $ \left(a^{2}+b^{2}\right) $ is equal to:

117.

Let $ L $ be a tangent to the parabola $ y^{2} = 4x - 20 $ at $ (6, 2) $. If $ L $ is also a tangent to the ellipse $ \dfrac{x^{2}}{2} + \dfrac{y^{2}}{b} = 1 $, then the value of $ b $ is equal to:

118.

The axis of a parabola is the line $ y = x $ and its vertex and focus are in the first quadrant at distances $ \sqrt{2} $ and $ 2\sqrt{2} $ units from the origin, respectively. If the point $ (1,k) $ lies on the parabola, then a possible value of $ k $ is:

119.

Consider a hyperbola $ H : {x^{2}} - {2y^{2}} = 4 $. Let the tangent at a point $ P(4,\sqrt{6}) $ meet the x-axis at $ Q $ and latus rectum at $ R(x_{1},y_{1}), x_{1} > 0 $. If $ F $ is a focus of $ H $ which is nearer to the point $ P $, then the area of $ \triangle QFR $ is equal to:

120.

Let $ f(x)=x^{2}+9 $, $ g(x)=\dfrac{x}{x-9} $ and $ a=f o g(10),\, b=g o f(3) $. If $ e $ and $ l $ denote the eccentricity and the length of the latus rectum of the ellipse $ \dfrac{x^{2}}{a}+\dfrac{y^{2}}{b}=1 $, then $ 8e^{2}+l^{2} $ is equal to:

121.

Let an ellipse $ E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\ a^{2}>b^{2} $, passes through $ \left(\dfrac{\sqrt{3}}{2},1\right) $ and has eccentricity $ \dfrac{1}{\sqrt{3}} $. If a circle, centered at focus $ F(\alpha,0),\ \alpha>0 $ of $ E $ and radius $ \dfrac{2}{\sqrt{3}} $ intersects $ E $ at two points $ P $ and $ Q $, then $ PQ^{2} $ is equal to:

122.

Let the line $ y - x = 1 $ intersect the ellipse $ \frac{x^{2}}{2} + \frac{y^{2}}{1} = 1 $ at the points $ A $ and $ B $. Then the angle made by the line segment $ AB $ at the center of the ellipse is:

123.

Two tangent lines $l_1$ and $l_2$ are drawn from the point $ (2, 0) $ to the parabola $ 2y^2 = -x $. If the lines $l_1$ and $l_2$ are also tangent to the circle $ (x-5)^2 + y^2 = r $, then $ 17r $ is equal to _______________.

124.

Let $ P $ be a point on the ellipse $\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1 $. Let the line passing through $ P $ and parallel to $ y $-axis meet the circle $ x^2 + y^2 = 9 $ at point $ Q $ such that $ P $ and $ Q $ are on the same side of the $ x $-axis. Then, the eccentricity of the locus of the point $ R $ on $ PQ $ such that $ PR:RQ = 4:3 $ as $ P $ moves on the ellipse, is:

125.

The radius of the smallest circle which touches the parabolas $ y = x^{2} + 2 $ and $ x = y^{2} + 2 $ is:

126.

If $ m $ is the slope of a common tangent to the curves $ \dfrac{x^{2}}{16} + \dfrac{y^{2}}{9} = 1 $ and $ x^{2} + y^{2} = 12 $, then $ 12 m^{2} $ is equal to:

127.

The area of the region enclosed by the curves $ y = x^{2} - 4x + 4 $ and $ y^{2} = 16 - 8x $ is:

128.

Let the hyperbola $ H : \frac{x^{2}}{a^{2}} - y^{2} = 1 $ and the ellipse $ E : 3x^{2} + 4y^{2} = 12 $ be such that the length of latus rectum of $ H $ is equal to the length of latus rectum of $ E $. If $ e_{H} $ and $ e_{E} $ are the eccentricities of $ H $ and $ E $ respectively, then the value of $ 12\left(e_{H}^{2} + e_{E}^{2}\right) $ is equal to ________ .

129.

Let $ P $ be a moving point on the circle $ x^2 + y^2 - 6x - 8y + 21 = 0 $. Then, the maximum distance of $ P $ from the vertex of the parabola $ x^2 + 6x + y + 13 = 0 $ is equal to:

130.

If the equation of the parabola, whose vertex is at $ (5,4) $ and the directrix is $ 3x + y - 29 = 0 $, is $ x^{2} + a y^{2} + bxy + cx + dy + k = 0 $, then $ a + b + c + d + k $ is equal to

131.

The area (in sq. units) of the region enclosed between the parabola $ y^{2} = 2x $ and the line $ x + y = 4 $ is ________ .

132.

Consider the sets $ A = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^{2} + y^{2} = 25\} $, $ B = \{(x,y) \in \mathbb{R} \times \mathbb{R} : x^{2} + 9y^{2} = 144\} $, $ C = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : x^{2} + y^{2} \le 4\} $ and $ D = A \cap B $. The total number of one-one functions from the set $ D $ to the set $ C $ is:

133.

If the maximum distance of normal to the ellipse $ \frac{x^2}{4} + \frac{y^2}{b^2} = 1, \; b < 2 $, from the origin is $ 1 $, then the eccentricity of the ellipse is:

134.

If the line $\alpha x+4y=\sqrt{7} $, where $ a\in \mathbb{R} $, touches the ellipse $ 3x^2+4y^2=1 $ at the point $ P $ in the first quadrant, then one of the focal distances of $ P $ is:

135.

Let $ l $ be a line which is normal to the curve $ y = 2x^2 + x + 2 $ at a point $ P $ on the curve. If the point $ Q(6, 4) $ lies on the line $ l $ and $ O $ is origin, then the area of the triangle $ OPQ $ is equal to ________.

136.

Let $ S $ be the focus of the hyperbola $ \dfrac{x^{2}}{3}-\dfrac{y^{2}}{5}=1 $, on the positive $ x $-axis. Let $ C $ be the circle with its centre at $ A(\sqrt{6},\sqrt{5}) $ and passing through the point $ S $. If $ O $ is the origin and $ SAB $ is a diameter of $ C $, then the square of the area of the triangle $ OSB $ is equal to ________.

137.

Let $ \lambda x - 2y = \mu $ be a tangent to the hyperbola $ a^{2}x^{2} - y^{2} = b^{2} $. Then $ \left( \frac{\lambda}{a} \right)^{2} - \left( \frac{\mu}{b} \right)^{2} $ is equal to:

138.

For which of the following curves, the line $x+\sqrt{3}\,y=2\sqrt{3}$ is the tangent at the point $\left(\dfrac{3\sqrt{3}}{2},\dfrac{1}{2}\right)$?

139.

Let $ A(\alpha, 0) $ and $ B(0, \beta) $ be the points on the line. Let the point $ P $ divide the line segment $ AB $ internally in the ratio $ 7:3 $. Let $ 3x - 25 = 0 $ be a directrix of the ellipse $ E: \, \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 $ and the corresponding focus be $ S $. If from $ S $, the perpendicular on the $ x $-axis passes through $ P $, then the length of the latus rectum of $ E $ is equal to:

140.

Let the foci of the ellipse $ \dfrac{x^{2}}{16} + \dfrac{y^{2}}{7} = 1 $ and the hyperbola $ \dfrac{x^{2}}{144} - \dfrac{y^{2}}{\alpha} = \dfrac{1}{25} $ coincide. Then the length of the latus rectum of the hyperbola is:

141.

Let $ H:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1 , \ a>0 , \ b>0 $ be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $ 4\left(2\sqrt{2}+\sqrt{14}\right) $. If the eccentricity $ H $ is $ \dfrac{\sqrt{11}}{2} $, then the value of $ a^2+b^2 $ is equal to __________.

142.

Let chord $ PQ $ of length $ 3\sqrt{13} $ of the parabola $ y^2 = 12x $ be such that the ordinates of points $ P $ and $ Q $ are in the ratio $ 1:2 $. If the chord $ PQ $ subtends an angle $ \alpha $ at the focus of the parabola, then $ \sin \alpha $ is equal to:

143.

Let $ P(x_{0},y_{0}) $ be the point on the hyperbola $ 3x^{2}-4y^{2}=36 $, which is nearest to the line $ 3x+2y=1 $. Then $ \sqrt{2}\,(y_{0}-x_{0}) $ is equal to:

144.

Let a tangent to the curve $ 9x^{2} + 16y^{2} = 144 $ intersect the coordinate axes at the points $ A $ and $ B $. Then, the minimum length of the line segment $ AB $ is _______.

145.

A square $ ABCD $ has all its vertices on the curve $ x^2 y^2 = 1 $. The midpoints of its sides also lie on the same curve. Then, the square of area of $ ABCD $ is ______________ .

146.

Let a parabola $ P $ be such that its vertex and focus lie on the positive $ x $-axis at a distance $ 2 $ and $ 4 $ units from the origin, respectively. If tangents are drawn from $ O(0,0) $ to the parabola $ P $ which meet $ P $ at $ S $ and $ R $, then the area (in sq. units) of $ \triangle S O R $ is equal to:

147.

If the shortest distance of the parabola $ y^2 = 4x $ from the centre of the circle $ x^2 + y^2 - 4x - 16y + 64 = 0 $ is $ d $, then $ d^2 $ is equal to:

148.

Suppose $ AB $ is a focal chord of the parabola $ y^{2} = 12x $ of length $ l $ and slope $ m < \sqrt{3} $. If the distance of the chord $ AB $ from the origin is $ d $, then $ l d^{2} $ is equal to __________.

149.

Let the hyperbola $ H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ pass through the point $ (2\sqrt{2}, -2\sqrt{2}) $. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is $ e $ times the length of the latus rectum of H, where $ e $ is the eccentricity of H, then which of the following points lies on the parabola

150.

Let $ y=mx+c,\; m>0 $ be the focal chord of $ y^{2}=-64x $, which is tangent to $ (x+10)^{2}+y^{2}=4 $. Then, the value of $ 4\sqrt{2}\,(m+c) $ is equal to _______.

151.

Let $ R $ be a rectangle given by the lines $ x=0 $, $ x=2 $, $ y=0 $ and $ y=5 $. Let $ A(\alpha, 0) $ and $ B(0, \beta) $, $ \alpha \in [0, 2] $ and $ \beta \in [0, 5] $, be such that the line segment $ AB $ divides the area of the rectangle $ R $ in the ratio $ 4:1 $. Then, the mid-point of $ AB $ lies on a :

152.

The shortest distance between the curves $ y^2 = 8x $ and $ x^2 + y^2 + 12y + 35 = 0 $ is:

153.

Let $ C $ be the largest circle centred at $ (2, 0) $ and inscribed in the ellipse $ \frac{x^{2}}{36} + \frac{y^{2}}{16} = 1 $. If $ (1, \alpha) $ lies on $ C $, then $ 10\alpha^{2} $ is equal to ______.

154.

Let $ P: y^{2}=4ax,\; a>0 $ be a parabola with focus $ S $. Let the tangents to the parabola $ P $ make an angle of $ \frac{\pi}{4} $ with the line $ y=3x+5 $ touch the parabola $ P $ at $ A $ and $ B $. Then the value of $ a $ for which $ A $, $ B $ and $ S $ are collinear is:

155.

Let $ P(4, 4\sqrt{3}) $ be a point on the parabola $ y^2 = 4ax $ and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

156.

If $ y=m_{1}x+c_{1} $ and $ y=m_{2}x+c_{2},\ m_{1}\neq m_{2} $ are two common tangents of circle $ x^{2}+y^{2}=2 $ and parabola $ y^{2}=x $, then the value of $ 8|\,m_{1}m_{2}\,| $ is equal to:

157.

If the minimum area of the triangle formed by a tangent to the ellipse $ \dfrac{x^2}{b^2} + \dfrac{y^2}{4a^2} = 1 $ and the co-ordinate axis is $ kab $, then $ k $ is equal to $ \_\_\_\_\_\_\_\_\_\_ $.

158.

The ordinates of the points $ P $ and $ Q $ on the parabola with focus $ (3,0) $ and directrix $ x=-3 $ are in the ratio $ 3:1 $. If $ R(\alpha,\beta) $ is the point of intersection of the tangents to the parabola of $ P $ and $ Q $, then $ \dfrac{\beta^2}{\alpha} $ is equal to ________.

159.

Let the eccentricity of a hyperbola satisfy the equation $ 6e^{2} - 11e + 3 = 0 $. If the foci of the hyperbola are $ (3, 5) $ and $ (3, -4) $, then the length of its latus rectum is:

160.

Two parabolas have the same focus $ (4,3) $ and their directrices are the $ x $-axis and the $ y $-axis, respectively. If these parabolas intersects at the points $ A $ and $ B $, then $ (AB)^2 $ is equal to:

161.

Let a tangent be drawn to the ellipse $ \dfrac{x^{2}}{27} + y^{2} = 1 $ at $ \left(3\sqrt{3}\cos\theta,\ \sin\theta\right) $ where $ \theta \in \left(0,\ \dfrac{\pi}{2}\right) $. Then the value of $ \theta $ such that the sum of intercepts on axes made by this tangent is minimum is equal to:

162.

A tangent and a normal are drawn at the point $ P(2,\,-4) $ on the parabola $ y^2 = 8x $, which meet the directrix of the parabola at the points $ A $ and $ B $ respectively. If $ Q(a,\, b) $ is a point such that $ AQBP $ is square, then $ 2a + b $ is equal to:

163.

Let the equation of two diameters of a circle $ x^2 + y^2 - 2x + 2fy + 1 = 0 $ be $ 2px - y = 1 $ and $ 2x + py = 4p $. Then the slope $ m \in (0,\infty) $ of the tangent to the hyperbola $ 3x^2 - y^2 = 3 $ passing through the centre of the circle is equal to ________.

164.

Let an ellipse $ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 $, pass through the point (4, 3) and have eccentricity Then the length of its latus rectum is:

165.

Let for two distinct values of $ p $ the lines $ y = x + p $ touch the ellipse E: $ \frac{x^2}{4^2} + \frac{y^2}{3^2} = 1 $ at the points $ A $ and $ B $. Let the line $ y = x $ intersect $ E $ at the points $ C $ and $ D $. Then the area of the quadrilateral $ ABCD $ is equal to:

166.

Let the eccentricity of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, $ a > b $, be $ \frac{1}{4} $. If this ellipse passes through the point $ \left(-4\sqrt{\frac{2}{5}},\, 3\right) $, then $ a^2 + b^2 $ is equal to:

167.

Let $ (h,k) $ lie on the circle $ C:x^{2}+y^{2}=4 $ and the point $ (2h+1,\,3k+2) $ lie on an ellipse with eccentricity $ e $. Then the value of $ \dfrac{5}{e^{2}} $ is equal to ____________.

168.

If the area of the triangle formed by the positive $ x $-axis, the normal and the tangent to then circle $ (x-2)^2 + (y-3)^2 = 25 $ at the point $ (5,7) $ is $ A $, then $ 24A $ is equal to __________.

169.

The locus of mid-points of the line segments joining $ (-3,-5) $ and the points on the ellipse $ \frac{x^{2}}{4} + \frac{y^{2}}{9} = 1 $ is:

170.

Let $ H : \frac{-x^2}{a^2} + \frac{y^2}{b^2} = 1 $ be the hyperbola, whose eccentricity is $ \sqrt{3} $ and the length of the latus rectum is $ 4\sqrt{3} $. Suppose the point $ (\alpha, 6) $, $ \alpha > 0 $ lies on $ H $. If $ \beta $ is the product of the focal distances of the point $ (\alpha, 6) $, then $ \alpha^2 + \beta $ is equal to:

171.

Let the foci of a hyperbola $ H $ coincide with the foci of the ellipse $\displaystyle E:\ \frac{(x-1)^{2}}{100}+\frac{(y-1)^{2}}{75}=1 $ and the eccentricity of the hyperbola $ H $ be the reciprocal of the eccentricity of the ellipse $ E $. If the length of the transverse axis of $ H $ is $ \alpha $ and the length of its conjugate axis is $ \beta $, then $ 3\alpha^{2}+2\beta^{2} $ is equal to

172.

Let $ PQ $ be a focal chord of the parabola $ y^{2} = 36x $ of length $ 100 $, making an acute angle with the positive $ x $-axis. Let the ordinate of $ P $ be positive and $ M $ be the point on the line segment $ PQ $ such that $ PM : MQ = 3 : 1 $. Then which of the following points does NOT lie on the line passing through $ M $ and perpendicular to the line $ PQ $?

173.

Let $ a>0, b>0 $. Let $ e $ and $ l $ respectively be the eccentricity and length of the latus rectum of the hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $. Let $ e' $ and $ l' $ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $ e^2 = \dfrac{11}{14}\,l $ and $ (e')^2 = \dfrac{11}{8}\,l' $, then the value of $ 77a + 44b $ is equal to:

174.

The locus of the mid point of the line segment joining the point $ (4, 3) $ and the points on the ellipse $ x^2 + 2y^2 = 4 $ is an ellipse with eccentricity:

175.

Let the mirror image of a circle $ c_1:x^2+y^2-2x-6y+\alpha=0 $ in line $ y=x+1 $ be $ c_2:5x^2+5y^2+10gx+10fy+38=0 $. If $ r $ is the radius of circle $ c_2 $, then $ \alpha+6r^2 $ is equal to__________.

176.

Let $ C $ be the locus of the mirror image of a point on the parabola $ y^{2}=4x $ with respect to the line $ y=x $. Then the equation of tangent to $ C $ at $ P(2,1) $ is:

177.

Let $ \theta $ be the acute angle between the tangents to the ellipse $ \frac{x^2}{9} + \frac{y^2}{1} = 1 $ and the circle $ x^2 + y^2 = 3 $ at their point of intersection in the first quadrant. Then $ \tan \theta $ is equal to:

178.

The length of the chord of the ellipse $ \displaystyle\frac{x^2}{4} + \frac{y^2}{2} = 1 $, whose mid-point is $\displaystyle \left( 1, \frac{1}{2} \right) $, is:

179.

An equilateral triangle $ OAB $ is inscribed in the parabola $ y^2 = 4x $ with the vertex $ O $ at the vertex of the parabola. Then the minimum distance of the circle having $ AB $ as a diameter from the origin is:

180.

If the points of intersection of two distinct conics $ x^2 + y^2 = 4b $ and $ \dfrac{x^2}{16} + \dfrac{y^2}{b^2} = 1 $ lie on the curve $ y^2 = 3x^2 $, then $ 3\sqrt{3} $ times the area of the rectangle formed by intersection points is __________.

181.

Let the product of the focal distance of the point $ P(4, 2\sqrt{3}) $ on the hyperbola $ H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be $ 32 $. Let the length of the conjugate axis of $ H $ be $ p $ and the length of its latus rectum be $ q $. Then $ p^2 + q^2 $ is equal to ________.

182.

Let the maximum area of the triangle that can be inscribed in the ellipse $ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1 $, $ a > 2 $, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the $ y $-axis, be $ 6\sqrt{3} $. Then the eccentricity of the ellipse is:

183.

$\text{Consider the hyperbola } \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text{ having one of its focus at } P(-3,0). \text{ If the latus}$ $\text{rectum through its other focus subtends a right angle at } P \text{ and } a^2b^2 = \alpha\sqrt{2} - \beta, \alpha, \beta \in \mathbb{N},$ $\text{then } \alpha + \beta \text{ is ____________ .}$

184.

The line $ y = x + 1 $ meets the ellipse $ \frac{x^2}{4} + \frac{y^2}{2} = 1 $ at two points $ P $ and $ Q $. If $ r $ is the radius of the circle with $ PQ $ as diameter then $ (3r)^2 $ is equal to:

185.

The eccentricity of an ellipse $ E $ with centre at the origin $ O $ is $ \dfrac{\sqrt{3}}{2} $ and its directrices are $ x = \pm \dfrac{4\sqrt{6}}{3} $. Let $ H : \dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1 $ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $ E $, and whose length of latus rectum is equal to the length of minor axis of $ E $. Then the distance between the foci of $ H $ is:

186.

Let the tangent to the parabola $ S: y^{2}=2x $ at the point $ P(2,2) $ meet the $ x $-axis at $ Q $ and normal at it meet the parabola $ S $ at the point $ R $. Then the area (in sq. units) of the triangle $ PQR $ is equal to:

187.

Consider the parabola $ P: y^{2}=4kx $ and the ellipse $ E: \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 $. Let the line segment joining the points of intersection of $ P $ and $ E $, be their latus rectums. If the eccentricity of $ E $ is $ e $, then $ e^{2}+2\sqrt{2} $ is equal to ________.

188.

A circle of radius $ 2 $ unit passes through the vertex and the focus of the parabola $ y^{2} = 2x $ and touches the parabola $ y = \left( x - \dfrac{1}{4} \right)^{2} + \alpha $, where $ \alpha > 0 $. Then $ (4\alpha - 8)^{2} $ is equal to _______.

189.

Let $ S $ and $ S' $ be the foci of the ellipse $ \frac{x^2}{25} + \frac{y^2}{9} = 1 $ and $ P(\alpha, \beta) $ be a point on the ellipse in the first quadrant. If $ (SP)^2 + (S'P)^2 - SP \cdot S'P = 37 $, then $ \alpha^2 + \beta^2 $ is equal to:

190.

The equation of a common tangent to the parabolas $ y = x^2 $ and $ y = -(x-2)^2 $ is:

191.

Let $ S=\{(x,y)\in \mathbb{N}\times \mathbb{N}:9(x-3)^2+16(y-4)^2\leq 144\} $ and $\displaystyle T = \{(x, y) \in \mathbb{R} \times \mathbb{R} : (x - 7)^2 + (y - 4)^2 \le 36\}$. Then $n(S\cap T)$ is equal to __________.

192.

Let $ e_1 $ be the eccentricity of the hyperbola $\displaystyle \frac{x^2}{16}-\frac{y^2}{9}=1 $ and $ e_2 $ be the eccentricity of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b $, which passes through the foci of the hyperbola. If $ e_1 e_2=1 $, then the length of the chord of the ellipse parallel to the $ x $-axis and passing through $ (0,2) $ is:

193.

If a tangent to the ellipse $ {x^2} + 4y^2 = 4 $ meets the tangents at the extremities of its major axis at $ B $ and $ C $, then the circle with $ BC $ as diameter passes through the point:

194.

If the line $ \alpha x + 2y = 1 $, where $ \alpha \in \mathbb{R} $, does not meet the hyperbola $ x^2 - 9y^2 = 9 $, then a possible value of $ \alpha $ is:

195.

The point $ P(-2\sqrt{6}, \sqrt{3}) $ lies on the hyperbola $ \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 $ having eccentricity $ \dfrac{\sqrt{5}}{2} $. If the tangent and normal at $ P $ to the hyperbola intersect its conjugate axis at the points $ Q $ and $ R $ respectively, then $ QR $ is equal to:

196.

The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, -3)$ and eccentricity is $\frac{4}{5}$, is:

197.

A triangle is formed by the tangents at the point $ (2, 2) $ on the curves $ y^2 = 2x $ and $ x^2 + y^2 = 4x $, and the line $ x + y + 2 = 0 $. If $ r $ is the radius of its circumcircle, then $ r^2 $ is equal to ________.

198.

If vertex of a parabola is $ (2,-1) $ and the equation of its directrix is $ 4x - 3y = 21 $, then the length of its latus rectum is:

199.

Let the foci and length of the latus rectum of an ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \ a > b $ be $ (\pm 5, 0) $ and $ \sqrt{50} $, respectively. Then, the square of the eccentricity of the hyperbola $ \frac{x^2}{b^2} - \frac{y^2}{a^2 b^2} = 1 $ equals:

200.

The locus of the mid points of the chords of the hyperbola $ x^2 - y^2 = 4 $, which touch the parabola $ y^2 = 8x $, is :

201.

Let the eccentricity of the hyperbola $ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 $ be $ \frac{5}{4} $. If the equation of the normal at the point $ \left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right) $ on the hyperbola is $ 8\sqrt{5}\,x+\beta y=\lambda $, then $ \lambda-\beta $ is equal to __________.

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