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Let \( P \) be the parabola, whose focus is \( (-2,1) \) and directrix is \( 2x + y + 2 = 0 \). Then the sum of the ordinates of the points on \( P \), whose abscissa is \( -2 \), is:

Let the quadratic curve passing through the point \( (-1, 0) \) and touching the line \( y = x \) at \( (1, 1) \) be \( y = f(x) \). Then the x-intercept of the normal to the curve at the point \( (\alpha, \alpha + 1) \) in the first quadrant is ______.

If the midpoint of a chord of the ellipse \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \) is \( \left(\sqrt{2}, \frac{4}{3}\right) \), and the length of the chord is \( \frac{2\sqrt{a}}{3} \), then \( \alpha \) is:

Let \( H_1 : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \) and \( H_2 : -\dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1 \) be two hyperbolas having length of latus rectums \( 15\sqrt{2} \) and \( 12\sqrt{5} \) respectively. Let their eccentricities be \( e_1 = \sqrt{\dfrac{5}{2}} \) and \( e_2 \) respectively. If the product of the lengths of their transverse axes is \( 100\sqrt{10} \), then \( 25e_2^2 \) is equal to \( \underline{\phantom{xxxx}} \).

Let the length of the transverse and conjugate axes of a hyperbola in standard form be \( 2a \) and \( 2b \) respectively, and one focus and the corresponding directrix of this hyperbola be \( ( +5,0 ) \) and \( 5x+9=0 \), respectively. If the product of the focal distances of a point \( (u,2\sqrt{5}) \) on the hyperbola is \( p \), then \( 4p \) equal to __________.

Let \( m_1 \) and \( m_2 \) be the slopes of the tangents drawn from the point \( P(4, 1) \) to the hyperbola \( H: \dfrac{y^2}{25} - \dfrac{x^2}{16} = 1 \). If \( Q \) is the point from which the tangents drawn to \( H \) have slopes \( |m_1| \) and \( |m_2| \) and they make positive intercepts \( \alpha \) and \( \beta \) on the \( x \)-axis, then \( \dfrac{(PQ)^2}{\alpha \beta} \) is equal to ______.

Let \( e_1 \) and \( e_2 \) be the eccentricities of the ellipse \( \dfrac{x^{2}}{b^{2}} + \dfrac{y^{2}}{25} = 1 \) and the hyperbola \( \dfrac{x^{2}}{16} - \dfrac{y^{2}}{b^{2}} = 1 \), respectively. If \( b < 5 \) and \( e_1 e_2 = 1 \), then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

A straight line cuts off the intercepts \( OA = a \) and \( OB = b \) on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin \( O \) to this makes an angle of \( \frac{\pi}{6} \) with positive direction of y-axis and the area of \( \triangle OAB \) is \( \frac{98}{3}\sqrt{3} \), then \( a^{2}-b^{2} \) is equal to:

Let \( A \) and \( B \) be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^{2} = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between \( A \) and \( B \), and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^{2} = 4x \), then the value of \( (a + d) \) is ________.

If \( \alpha x+\beta y=109 \) is the equation of the chord of the ellipse \(\displaystyle \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \), whose mid point is \(\displaystyle \left(\frac{5}{2},\frac{1}{2}\right) \), then \( \alpha+\beta \) is equal to:

If \( P(h, k) \) be a point on the parabola \( x = 4y^{2} \), which is nearest to the point \( Q(0, 33) \), then the distance of \( P \) from the directrix of the parabola \( y^{2} = 4(x + y) \) is equal to:

Let the ellipse \( E:x^{2}+9y^{2}=9 \) intersect the positive \( x \)- and \( y \)-axes at the points \( A \) and \( B \) respectively. Let the major axis of \( E \) be a diameter of the circle \( C \). Let the line passing through \( A \) and \( B \) meet the circle \( C \) at the point \( P \). If the area of the triangle with vertices \( A \), \( P \) and the origin \( O \) is \( \dfrac{m}{n} \), where \( m \) and \( n \) are coprime, then \( m-n \) is equal to:

Let \( P(3\cos\alpha, 2\sin\alpha), \alpha \neq 0 \), be a point on the ellipse \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \), Q be a point on the circle \( x^2 + y^2 - 14x - 14y + 82 = 0 \) and R be a point on the line \( x + y = 5 \) such that the centroid of the triangle PQR is \( \left( 2 + \cos\alpha, 3 + \frac{2}{3}\sin\alpha \right) \). Then the sum of the ordinates of all possible points R is:

If the line \( y = 4 + kx, \; k > 0 \), is the tangent to the parabola \( y = x - x^2 \) at the point \( P \) and \( V \) is the vertex of the parabola, then the slope of the line through \( P \) and \( V \) is:

The equation of the chord, of the ellipse \(\displaystyle \frac{x^2}{25} + \frac{y^2}{16} = 1 \), whose mid-point is \( (3, 1) \) is:

The combined equation of the two lines \( ax+by+c=0 \) and \( a'x+b'y+c'=0 \) can be written as \( (ax+by+c)(a'x+b'y+c')=0 \). The equation of the angle bisectors of the lines represented by the equation \( 2x^{2}+xy-3y^{2}=0 \) is:

Let the system of linear equations \( x+y+kz=2 \), \( 2x+3y-z=1 \), \( 3x+4y+2z=k \) have infinitely many solutions. Then the system
\( (k+1)x+(2k-1)y=7 \)
\( (2k+1)x+(k+5)y=10 \)
has:

Let the length of a latus rectum of an ellipse \( \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \) be \( 10 \). If its eccentricity is the minimum value of the function \( f(t) = t^{2} + t + \dfrac{11}{12},\, t \in \mathbb{R} \), then \( a^{2} + b^{2} \) is equal to:

Let \( y=x+2 \), \( 4y=3x+6 \) and\( 3y=4x+1 \) be three tangent lines to the circle \( (x-h)^2+(y-k)^2=r^2 \). Then \( h+k \) is equal to:

Let \( A=\{(\alpha,\beta)\in R\times R :|\alpha-1|\le 4 \text{ and } |\beta-5|\le 6\} \) and \( B=\{(\alpha,\beta)\in R\times R:\ 16(\alpha-2)^{2}+9(\beta-6)^{2}\le 144\} \). Then

If \( A \) and \( B \) are the points of intersection of circle \( x^{2}+y^{2}-8x=0 \) and the hyperbola \( \frac{x^{2}}{9}-\frac{y^{2}}{4}=1 \) and a point \( P \) moves on the line \( 2x-3y+4=0 \), then the centroid of \( \triangle PAB \) lies on the line:

Let a common tangent to the curves \( y^{2}=4x \) and \( (x-4)^{2}+y^{2}=16 \) touch the curves at the points \( P \) and \( Q \). Then \( (PQ)^{2} \) is equal to ________.

Let \( y^{2}=12x \) be the parabola and \( S \) be its focus. Let \( PQ \) be a focal chord of the parabola such that \( (SP)(SQ)=\dfrac{147}{4} \). Let \( C \) be the circle described taking \( PQ \) as diameter. If the equation of a circle \( C \) is \( 64x^{2}+64y^{2}-\alpha x-64\sqrt{3}\,y=\beta \), then \( \beta-\alpha \) is equal to _______.

If the equation of the parabola with vertex \( V\left(\frac{3}{2},\,3\right) \) and the directrix \( x+2y=0 \) is \( \alpha x^{2}+\beta y^{2}-\gamma xy-30x-60y+225=0 \), then \( \alpha+\beta+\gamma \) is equal to:

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JEE Main 2023 1st Feb Morning Shift

The combined equation of the two lines \( ax+by+c=0 \) and \( a'x+b'y+c'=0 \) can be written as \( (ax+by+c)(a'x+b'y+c')=0 \). The equation of the angle bisectors of the lines represented by the equation \( 2x^{2}+xy-3y^{2}=0 \) is: