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JEE Mains | PYQ

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Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \dfrac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \dfrac{x^{2}}{\alpha^{2}} - \dfrac{y^{2}}{\beta^{2}} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2\alpha^{2} + 3\beta^{2} \) is equal to ________.

Two tangents are drawn from a point \( P \) to the circle \( x^{2}+y^{2}-2x-4y+4=0 \), such that the angle between these tangent is \( \tan^{-1}\!\left(\dfrac{12}{5}\right) \), where \( \tan^{-1}\!\left(\dfrac{12}{5}\right)\in(0,\pi) \). If the centre of the circle is denotes by \( C \) and these tangents touch the circle at points \( A \) and \( B \), then the ratio of the areas of \( \triangle PAB \) and \( \triangle CAB \) is:

Let \( C \) be the circle of minimum area touching the parabola \( y = 6 - x^2 \) and the lines \( y = \sqrt{3}\,|x| \). Then, which one of the following points lies on the circle \( C \)?

A circle passing through the point \( P(\alpha, \beta) \) in the first quadrant touches the two coordinate axes at the points \( A \) and \( B \). The point \( P \) is above the line \( AB \). The point \( Q \) on the line segment \( AB \) is the foot of perpendicular from \( P \) on \( AB \). If \( PQ \) is equal to \( 11 \) units, then the value of \( \alpha \beta \) is ______________.

Let \( C \) be a circle having centre in the first quadrant and touching the \( x \)-axis at a distance of \( 3 \) units from the origin. If the circle \( C \) has an intercept of length \( 6\sqrt{3} \) on \( y \)-axis, then the length of the chord of the circle \( C \) on the line \( x - y = 3 \) is:

Four distinct points \( (2k,3k) \), \( (1,0) \), \( (0,1) \) and \( (0,0) \) lie on a circle for \( k \) equal to:

Let \( C_{1} \) be the circle in the third quadrant of radius \( 3 \), that touches both coordinate axes. Let \( C_{2} \) be the circle with centre \( (1,3) \) that touches \( C_{1} \) externally at the point \( (\alpha,\beta) \). If \( (\beta - \alpha)^{2} = \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to:

The sum of diameters of the circles that touch (i) the parabola \( 75x^2 = 64(5y - 3) \) at the point \( \left(\dfrac{8}{5}, \dfrac{6}{5}\right) \) and (ii) the y-axis, is equal to ________.

Let a circle passing through \( (2,0) \) have its centre at the point \( (h,k) \). Let \( (x_c,y_c) \) be the point of intersection of the lines \( 3x + 5y = 1 \) and \( (2 + c)x + 5c^2 y = 1 \). If \( h = \ \displaystyle \lim_{c \to 1} X_c and \ k = \lim_{c \to 1} y_c \), then the equation of the circle is:

The line \( 2x - y + 1 = 0 \) is a tangent to the circle at the point \( (2, 5) \) and the centre of the circle lies on \( x - 2y = 4 \). Then, the radius of the circle is:

Let the point \( (p, \, p + 1) \) lie inside the region \( E = \{ (x, y) : 3 - x \le y \le \sqrt{9 - x^2}, \, 0 \le x \le 3 \} \). If the set of all values of \( p \) is the interval \( (a, b) \), then \( b^2 + b - a^2 \) is equal to ______________.

Let \( y = x \) be the equation of a chord of the circle \( C_{1} \) (in the closed half-plane \( x \ge 0 \)) of diameter \( 10 \) passing through the origin. Let \( C_{2} \) be another circle described on the given chord as its diameter. If the equation of the chord of the circle \( C_{2} \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_{2} \), is \( x + a y + b = 0 \), then \( a - b \) is equal to:

Let \( r_{1} \) and \( r_{2} \) be the radii of the largest and smallest circles, respectively, which pass through the point \( (-4,1) \) and have their centres on the circumference of the circle \( x^{2}+y^{2}+2x+4y-4=0 \). If \( \dfrac{r_{1}}{r_{2}}=a+b\sqrt{2} \), then \( a+b \) is equal to:

Consider a circle \( (x-\alpha)^{2} + (y-\beta)^{2} = 50 \), where \( \alpha, \beta > 0 \). If the circle touches the line \( y + x = 0 \) at the point \( P \), whose distance from the origin is \( 4\sqrt{2} \), then \( (\alpha + \beta)^{2} \) is equal to __________.

Equations of two diameters of a circle are \( 2x - 3y = 5 \) and \( 3x - 4y = 7 \). The line joining the points \( \left(-\frac{22}{7}, -4 \right) \) and \( \left( -\frac{1}{7}, 3 \right) \) intersects the circle at only one point \( P(\alpha,\beta) \). Then \( 17\beta - \alpha \) is equal to ________.

Let a circle \( C_{1} \) be obtained on rolling the circle \( x^{2}+y^{2}-4x-6y+11=0 \) upwards \( 4 \) units on the tangent \( T \) to it at the point \( (3,2) \). Let \( C_{2} \) be the image of \( C_{1} \) in \( T \). Let \( A \) and \( B \) be the center of circles \( C_{1} \) and \( C_{2} \) respectively, and \( M \) and \( N \) be respectively the feet of perpendiculars drawn from \( A \) and \( B \) on the \( x \)-axis. Then the area of trapezium \( AMNB \) is:

The minimum distance between any two points \( P_{1} \) and \( P_{2} \) while considering point \( P_{1} \) on one circle and point \( P_{2} \) on the other circle for the given circles' equations \( x^{2} + y^{2} - 10x - 10y + 41 = 0 \) and \( x^{2} + y^{2} - 24x - 10y + 160 = 0 \) is ______________.

Let the lengths of intercepts on x-axis and y-axis made by the circle \( x^2 + y^2 + ax + 2ay + c = 0, \ (a < 0) \) be \( 2\sqrt{2} \) and \( 2\sqrt{5} \), respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line \( x + 2y = 0 \), is equal to:

Let the circle \( C_{1}: x^{2}+y^{2}-2(x+y)+1=0 \) and \( C_{2} \) be a circle having centre at \( (-1,0) \) and radius \( 2 \). If the line of the common chord of \( C_{1} \) and \( C_{2} \) intersects the y-axis at the point \( P \), then the square of the distance of \( P \) from the centre of \( C_{1} \) is:

The set of values of \( k \), for which the circle \( C:\; 4x^2 + 4y^2 - 12x + 8y + k = 0 \) lies inside the \( 4^{\text{th}} \) quadrant and the point \( \left(1, -\frac{1}{3}\right) \) lies on or inside the circle \( C \), is

Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is:

Let \( \mathbb{Z} \) be the set of all integers, \( A = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : (x-2)^{2} + y^{2} \leq 4 \} \), \( B = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : x^{2} + y^{2} \leq 4 \} \) and \( C = \{(x,y) \in \mathbb{Z} \times \mathbb{Z} : (x-2)^{2} + (y-2)^{2} \leq 4 \} \). If the total number of relations from \( A \cap B \) to \( A \cap C \) is \( 2^{p} \), then the value of \( p \) is :

Let \( B \) be the centre of the circle \( x^{2}+y^{2}-2x+4y+1=0 \). Let the tangents at two points \( P \) and \( Q \) on the circle intersect at the point \( A(3,1) \). Then \( 8\left(\dfrac{\text{area } \triangle APQ}{\text{area } \triangle BPQ}\right) \) is equal to ______.

Suppose that two chords, drawn from the point \( (1, 2) \) on the circle \( x^{2}+y^{2}+x-3y=0 \) are bisected by the \( y \)-axis. If the other ends of these chords are \( R \) and \( S \), and the midpoint of the line segment \( RS \) is \( (\alpha,\beta) \), then \( 6(\alpha+\beta) \) is equal to:

If the points of intersection of the ellipses \( x^2 + 2y^2 - 6x - 12y + 23 = 0 \) and \( 4x^2 + 2y^2 - 20x - 12y + 35 = 0 \) lie on a circle of radius \( r \) and centre \( (a,b) \), then the value of \( ab + 18r^2 \) is:

Let the parabola \( y = x^2 + px - 3 \), meet the coordinate axes at the points \( P, Q \) and \( R \). If the circle \( C \) with centre at \( (-1, -1) \) passes through the points \( P, Q \) and \( R \), then the area of \( \triangle PQR \) is:

Choose the correct statement about two circles whose equations are given below:
\( x^2 + y^2 - 10x - 10y + 41 = 0 \)
\( x^2 + y^2 - 22x - 10y + 137 = 0 \)

The points of intersection of the line \( ax + by = 0, \; (a \neq b) \) and the circle \( x^2 + y^2 - 2x = 0 \) are \( A(\alpha, 0) \) and \( B(1, \beta) \). The image of the circle with \( AB \) as a diameter in the line \( x + y + 2 = 0 \) is:

A rectangle \( R \) with end points of one of its sides as \( (1, 2) \) and \( (3, 6) \) is inscribed in a circle. If the equation of diameter of the circle is \( 2x - y + 4 = 0 \), then the area of \( R \) is __________.

Let a variable line passing through the centre of the circle \( x^{2}+y^{2}-16x-4y=0 \), meet the positive co-ordinate axes at the points \( A \) and \( B \). Then the minimum value of \( OA+OB \), where \( O \) is the origin, is equal to:

A line segment \( AB \) of length \( \lambda \) moves such that the points \( A \) and \( B \) remain on the periphery of a circle of radius \( \lambda \). Then the locus of the point, that divides the line segment \( AB \) in the ratio \( 2:3 \), is a circle of radius.

Let \( S_{1} : x^{2} + y^{2} = 9 \) and \( S_{2} : (x - 2)^{2} + y^{2} = 1 \). Then the locus of center of a variable circle \( S \) which touches \( S_{1} \) internally and \( S_{2} \) externally always passes through the points:

Let \( C \) be the centre of the circle \( x^2+y^2-x+2y=\dfrac{11}{4} \) and \( P \) be a point on the circle. A line passes through the point \( C \), makes an angle of \( \dfrac{\pi}{4} \) with the line \( CP \) and intersects the circle at the points \( Q \) and \( R \). Then the area of the triangle \( PQR \) (in unit\( ^2 \)) is:

Let \( P \) and \( Q \) be two distinct points on a circle which has centre at \( C(2,3) \) and which passes through origin \( O \). If \( OC \) is perpendicular to both the line segments \( CP \) and \( CQ \), then the set \( \{P, Q\} \) is equal to:

Let \( A = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid 2x^2 + 2y^2 - 2x - 2y = 1\} \), \( B = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid 4x^2 + 4y^2 - 16y + 7 = 0\} \) and \( C = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x^2 + y^2 - 4x - 2y + 5 \le r^2\} \}\).
Then the minimum value of \( |r| \) such that \( A \cup B \subseteq C \) is equal to:

Let \( PQ \) and \( MN \) be two straight lines touching the circle \( x^2 + y^2 - 4x - 6y - 3 = 0 \) at the points \( A \) and \( B \) respectively.
Let \( O \) be the centre of the circle and \( \angle AOB = \frac{\pi}{3} \). Then the locus of the point of intersection of the lines \( PQ \) and \( MN \) is:

The equation of a circle is \( \operatorname{Re}(z^{2}) + 2\,\operatorname({Im}(z))^{2} + 2\,\operatorname{Re}(z) = 0 \), where \( z = x + iy \). A line which passes through the centre of the given circle and the vertex of the parabola, \( x^{2} - 6x - y + 13 = 0 \), has \( y \) intercept equal to _______.

If the tangents at the points \( P \) and \( Q \) on the circle \( x^{2}+y^{2}-2x+y=5 \) meet at the point \( R\!\left(\dfrac{9}{4},\,2\right) \), then the area of the triangle \( PQR \) is:

The area enclosed by the closed curve \( C \) given by the differential equation \( \dfrac{dy}{dx}+\dfrac{x+a}{y-a}=0,\ y(1)=0 \) is \( 4\pi \).
Let \( P \) and \( Q \) be the points of interaction of the curve \( C \) and the y-axis. If normal at \( P \) and \( Q \) on the curve \( C \) intersect x-axis at points \( R \) and \( S \) respectively, then the length of the line segment \( RS \) is :

The absolute difference between the squares of the radii of the two circles passing through the point \( (-9,\,4) \) and touching the lines \( x+y=3 \) and \( x-y=3 \), is equal to __________.

Let the centre of a circle \( C \) be \( (\alpha,\beta) \) and its radius \( r<8 \). Let \( 3x+4y=24 \) and \( 3x-4y=32 \) be two tangents and \( 4x+3y=1 \) be a normal to \( C \). Then \( (\alpha-\beta+r) \) is equal to:

Let \( A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \} \), \( B = \{ z \in \mathbb{C} : \operatorname{Re}(z - iz) = 2 \} \) and \( S = A \cap B \). Then \( \sum\limits_{z \in S} |z|^{2} \) is equal to ______.

If one the diameters of the circle \( x^{2} + y^{2} - 2\sqrt{2}\,x - 6\sqrt{2}\,y + 14 = 0 \) is a chord of the circle \( (x - 2\sqrt{2})^{2} + (y - 2\sqrt{2})^{2} = r^{2} \), then the value of \( r^{2} \) is equal to \( \_ \_ \_ \_ \_ \_ \).

Let the circles \( C_1 : (x-\alpha)^2 + (y-\beta)^2 = r_1^2 \) and \( C_2 : (x-8)^2 + \left(y-\frac{15}{2}\right)^2 = r_2^2 \) touch each other externally at the point \( (6,6) \). If the point \( (6,6) \) divides the line segment joining the centres of the circles \( C_1 \) and \( C_2 \) internally in the ratio \( 2:1 \), then \( (\alpha+\beta) + 4(r_1^2 + r_2^2) \) equals:

Let a circle \( C \) touch the lines \( L_1: 4x-3y+K_1=0 \) and \( L_2: 4x-3y+K_2=0 \), \( K_1,K_2\in \mathbb{R} \). If a line passing through the centre of the circle \( C \) intersects \( L_1 \) at \( (-1,2) \) and \( L_2 \) at \( (3,-6) \), then the equation of the circle \( C \) is:

Let \( A \) be a point on the \( x \)-axis. Common tangents are drawn from \( A \) to the curves \( x^{2}+y^{2}=8 \) and \( y^{2}=16x \). If one of these tangents touches the two curves at \( Q \) and \( R \), then \( (QR)^{2} \) is equal to :

Consider a circle \( C \) which touches the y-axis at \( (0, 6) \) and cuts off an intercept \( 6\sqrt{5} \) on the x-axis. Then the radius of the circle \( C \) is equal to:

Let circle \( C \) be the image of \( x^2 + y^2 - 2x + 4y - 4 = 0 \) in the line \( 2x - 3y + 5 = 0 \) and \( A \) be the point on \( C \) such that \( OA \) is parallel to \( x \)-axis and \( A \) lies on the right hand side of the centre \( O \) of \( C \). If \( B\,(\alpha,\beta) \), with \( \beta < 4 \), lies on \( C \) such that the length of the arc \( AB \) is \( \left(\tfrac{1}{6}\right)^{\text{th}} \) of the perimeter of \( C \), then \( \beta - \sqrt{3}\alpha \) is equal to:

Two circles in the first quadrant of radii \( r_{1} \) and \( r_{2} \) touch the coordinate axes. Each of them cuts off an intercept of \( 2 \) units with the line \( x+y=2 \). Then \( r_{1}^{2}+r_{2}^{2}-r_{1}r_{2} \) is equal to _____.

Let the tangent to the circle \( C_1 : x^2 + y^2 = 2 \) at the point \( M(-1, 1) \) intersect the circle \( C_2 : (x - 3)^2 + (y - 2)^2 = 5 \), at two distinct points \( A \) and \( B \). If the tangents to \( C_2 \) at the points \( A \) and \( B \) intersect at \( N \), then the area of the triangle \( ANB \) is equal to:

If one of the diameters of the circle \( x^{2}+y^{2}-10x+4y+13=0 \) is a chord of another circle \( C \), whose centre is the point of intersection of the lines \( 2x+3y=12 \) and \( 3x-2y=5 \), then the radius of the circle \( C \) is:

For the four circles \( M \), \( N \), \( O \) and \( P \), following four equations are given:
Circle \( M \): \( x^2 + y^2 = 1 \)
Circle \( N \): \( x^2 + y^2 - 2x = 0 \)
Circle \( O \): \( x^2 + y^2 - 2x - 2y + 1 = 0 \)
Circle \( P \): \( x^2 + y^2 - 2y = 0 \)
If the centre of circle \( M \) is joined with centre of the circle \( N \), further centre of circle \( N \) is joined with centre of the circle \( O \), centre of circle \( O \) is joined with the centre of circle \( P \) and lastly, centre of circle \( P \) is joined with centre of circle \( M \), then these lines form the sides of a:

Let the circle \( S: 36x^2 + 36y^2 - 108x + 120y + c = 0 \) be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, \( x - 2y = 4 \) and \( 2x - y = 5 \) lies inside the circle \( S \), then:

Let \( C \) be a circle with radius \( \sqrt{10} \) units and centre at the origin. Let the line \( x + y = 2 \) intersects the circle \( C \) at the points \( P \) and \( Q \). Let \( MN \) be a chord of \( C \) of length \( 2 \) unit and slope \( -1 \). Then, a distance (in units) between the chord \( PQ \) and the chord \( MN \) is:

Let the centre of a circle, passing through the points \( (0,0) \), \( (1,0) \) and touching the circle \( x^2 + y^2 = 9 \), be \( (h,k) \). Then for all possible values of the coordinates of the centre \( (h,k) \), \( 4(h^2 + k^2) \) is equal to ________.

If the circles \( (x + 1)^{2} + (y + 2)^{2} = r^{2} \) and \( x^{2} + y^{2} - 4x - 4y + 4 = 0 \) intersect at exactly two distinct points, then:

For \( t \in (0, 2\pi) \), if \( ABC \) is an equilateral triangle with vertices \( A(\sin t, -\cos t) \), \( B(\cos t, \sin t) \) and \( C(a, b) \) such that its orthocenter lies on a circle with centre \( \left(1, \dfrac{1}{3}\right) \), then \( (a^2 - b^2) \) is equal to:

Let the tangent to the circle \( x^{2} + y^{2} = 25 \) at the point \( R(3,4) \) meet \( x \)-axis and \( y \)-axis at points \( P \) and \( Q \), respectively. If \( r \) is the radius of the circle passing through the origin \( O \) and having centre at the incentre of the triangle \( OPQ \), then \( r^{2} \) is equal to:

Let the maximum and minimum values of \( \, (\sqrt{8x - x^{2} - 12} - 4)^{2} + (x - 7)^{2} \, \), \( x \in \mathbb{R} \) be \( M \) and \( m \), respectively. Then \( M^{2} - m^{2} \) is equal to \( \_\_\_\_\_\_ \).

An arc \( PQ \) of a circle subtends a right angle at its centre \( O \). The midpoint of the arc \( PQ \) is \( R \). If \(\overrightarrow{ OP} =\vec{u} \), \(\overrightarrow{ OR} = \vec{v} \) and \(\overrightarrow{ OQ} = \ {\alpha \vec{u}} + \beta \vec{v} \), then \( \alpha, \ \beta^2 \) are the roots of equation.

Let the line \( x+y=1 \) meet the circle \( x^2+y^2=4 \) at the points \( A \) and \( B \). If the line perpendicular to \( AB \) and passing through the mid point of the chord \( AB \) intersects the circle at \( C \) and \( D \), then the area of the quadrilateral \( ADBC \) is equal to:

Let a triangle \( ABC \) be inscribed in the circle \( x^{2}-\sqrt{2}(x+y)+y^{2}=0 \) such that \( \angle BAC=\frac{\pi}{2} \). If the length of side \( AB \) is \( \sqrt{2} \), then the area of the \( \triangle ABC \) is equal to:

Let \( r \) be the radius of the circle, which touches \( x \)-axis at point \( (a,0) \), \( a<0 \) and the parabola \( y^{2}=9x \) at the point \( (4,6) \). Then \( r \) is equal to ______.

Let the tangents at two points A and B on the circle \( x^2 + y^2 - 4x + 3 = 0 \) meet at origin \( O(0, 0) \). Then the area of the triangle OAB is:

A square is inscribed in the circle \( x^2 + y^2 - 10x - 6y + 30 = 0 \). One side of this square is parallel to \( y = x + 3 \). If \( (x_i, y_i) \) are the vertices of the square, then \( \sum (x_i^2 + y_i^2 )\) is equal to:

Let \( C: x^{2}+y^{2}=4 \) and \( C' : x^{2}+y^{2}-4\lambda x+9=0 \) be two circles. If the set of all values of \( \lambda \) so that the circles \( C \) and \( C' \) intersect at two distinct points, is \( \mathbb{R}-[a,b] \), then the point \( (8a+12,\,16b-20) \) lies on the curve:

Let \( P(a_{1},b_{1}) \) and \( Q(a_{2},b_{2}) \) be two distinct points on a circle with center \( C(\sqrt{2},\sqrt{3}) \). Let \( O \) be the origin and \( OC \) be perpendicular to both \( CP \) and \( CQ \). If the area of the triangle \( OCP \) is \( \dfrac{\sqrt{35}}{2} \), then \( a_{1}^{2}+a_{2}^{2}+b_{1}^{2}+b_{2}^{2} \) is equal to __________.

Let \( x^{2} + y^{2} + Ax + By + C = 0 \) be a circle passing through \( (0, 6) \) and touching the parabola \( y = x^{2} \) at \( (2, 4) \). Then \( A + C \) is equal to :

A circle \( C \) touches the line \( x = 2y \) at the point \( (2, 1) \) and intersects the circle \( C_1 : x^2 + y^2 + 2y - 5 = 0 \) at two points \( P \) and \( Q \) such that \( PQ \) is a diameter of \( C_1 \). Then the diameter of \( C \) is:

Let \( C \) be a circle passing through the points \( A(2,-1) \) and \( B(3,4) \). The line segment \( AB \) is not a diameter of \( C \). If \( r \) is the radius of \( C \) and its centre lies on the circle \( (x-5)^{2}+(y-1)^{2}=\dfrac{13}{2} \), then \( r^{2} \) is equal to:

If the circle \( x^2 + y^2 - 2gx + 6y - 19c = 0, \; g, c \in \mathbb{R} \) passes through the point \( (6, 1) \) and its centre lies on the line \( x - 2cy = 8 \), then the length of intercept made by the circle on x-axis is:

Let \( A \) be the point \( (1, 2) \) and \( B \) be any point on the curve \( x^2 + y^2 = 16 \). If the centre of the locus of the point \( P \), which divides the line segment \( AB \) in the ratio \( 3 : 2 \) is the point \( C(\alpha, \beta) \), then the length of the line segment \( AC \) is:

Let the centre of the circle \( x^2+y^2+2gx+2fy+25=0 \) be in the first quadrant and lie on the line \( 2x-y=4 \). Let the area of an equilateral triangle inscribed in the circle be \( 27\sqrt{3} \). Then the square of the length of the chord of the circle on the line \( x=1 \) is ______.

Let the equation of the circle, which touches \( x \)-axis at the point \( (a,0) \), \( a>0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^{2}+y^{2}-\alpha x+\beta y+\gamma=0 \). If the circle lies below \( x \)-axis, the ordered pair \( (2a, b^{2}) \) is equal to:

If the tangents drawn at the points \( O(0,0) \) and \( P(1+\sqrt{5}, 2) \) on the circle \( x^2 + y^2 - 2x - 4y = 0 \) intersect at the point \( Q \), then the area of the triangle \( OPQ \) is equal to:

Let the equation \( x^2 + y^2 + p x + (1 - p) y + 5 = 0 \) represent circles of varying radius \( r \in (0, 5] \). Then the number of elements in the set \( S = \{ q : q = p^2 \text{ and } q \text{ is an integer} \} \) is ____________.

A circle touches both the y-axis and the line \( x + y = 0 \). Then the locus of its center is:

The set of all values of \( a^2 \) for which the line \( x+y=0 \) bisects two distinct chords drawn from a point \( P\!\left(\frac{1+a}{2},\frac{1-a}{2}\right) \) on the circle \( 2x^2+2y^2-(1+a)x-(1-a)y=0 \), is equal to:

The locus of a point, which moves such that the sum of squares of its distances from the points \( (0,0) \), \( (1,0) \), \( (0,1) \), \( (1,1) \) is \( 18 \) units, is a circle of diameter \( d \). Then \( d^{2} \) is equal to ________.

Two circles each of radius \( 5 \) units touch each other at the point \( (1, 2) \). If the equation of their common tangents is \( 4x + 3y = 10 \), and \( C_{1} (\alpha, \beta) \) and \( C_{2} (\gamma, \delta) \), \( C_{1} \ne C_{2} \) are their centres, then \( |(\alpha + \beta)(\gamma + \delta)| \) is equal to ________.

Let the abscissae of the two point \( P \) and \( Q \) on a circle be the roots of \( x^2 - 4x - 6 = 0 \) and the ordinates of \( P \) and \( Q \) be the roots of \( y^2 + 2y - 7 = 0 \). If \( PQ \) is a diameter of the circle \( x^2 + y^2 + 2ax + 2by + c = 0 \), then the value of \( (a + b - c) \) is ______.

If the four distinct points \( (4,6) \), \( (-1,5) \), \( (0,0) \) and \( (k,3k) \) lie on a circle of radius \( r \), then \( 10k + r^2 \) is equal to:

Let a focus of the ellipse \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) be \( S(4, 0) \) and its eccentricity be \( \frac{4}{5} \). If the point \( P(3, \alpha) \) lies on \( E \) and \( O \) is the origin, then the area of \( \triangle POS \) is equal to:

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines \( x + (k-1)y + 3 = 0 \) and \( 2x + k^2 y - 4 = 0 \). If the line \( x - y + 2 = 0 \) intersects the circle at the points \( A \) and \( B \), then \( (AB)^2 \) is equal to:

Let \( O \) be the origin and \( OP \) and \( OQ \) be the tangents to the circle \( x^{2} + y^{2} - 6x + 4y + 8 = 0 \) at the points \( P \) and \( Q \) on it. If the circumcircle of the triangle \( OPQ \) passes through the point \( \left(\alpha, \dfrac{1}{2}\right) \), then a value of \( \alpha \) is:

Let a point \( P \) be such that its distance from the point \( (5,0) \) is thrice the distance of \( P \) from the point \( (-5,0) \). If the locus of the point \( P \) is a circle of radius \( r \), then \( 4r^2 \) is equal to __________.

Let the abscissae of the two points \( P \) and \( Q \) be the roots of \( 2x^{2} - r x + p = 0 \) and the ordinates of \( P \) and \( Q \) be the roots of \( x^{2} - s x - q = 0 \). If the equation of the circle described on \( PQ \) as diameter is \( 2(x^{2} + y^{2}) - 11x - 14y - 22 = 0 \), then \( 2r + s - 2q + p \) is equal to ______.

Points \( P(-3,2), Q(9,10) \) and \( R(\alpha,4) \) lie on a circle \( C \) with \( PR \) as its diameter. The tangents to \( C \) at the points \( Q \) and \( R \) intersect at the point \( S \). If \( S \) lies on the line \( 2x - ky = 1 \), then \( k \) is equal to:

The acute angle between the pair of tangents drawn to the ellipse \( 2x^2+3y^2=5 \) from the point \( (1,3) \) is:

Let \( P \) and \( Q \) be any points on the curves \( (x-1)^2+(y+1)^2=1 \) and \( y=x^2 \), respectively. The distance between \( P \) and \( Q \) is minimum for some value of the abscissa of \( P \) in the interval:

If the circles \( x^{2}+y^{2}+6x+8y+16=0 \) and \( x^{2}+y^{2}+2(3-\sqrt{3})x+2(4-\sqrt{6})y = k+6\sqrt{3}+8\sqrt{6}, \, k>0 \), touch internally at the point \( P(\alpha,\beta) \), then \( (\alpha+\sqrt{3})^{2}+(\beta+\sqrt{6})^{2} \) is equal to ____________ .

The locus of the mid points of the chords of the circle \( C_1 : (x-4)^2 + (y-5)^2 = 4 \) which subtend an angle \( \theta_i \) at the centre of the circle \( C_1 \), is a circle radius \( r_i \). If \( \theta_1 = \frac{\pi}{3} \), \( \theta_3 = \frac{2\pi}{3} \) and \( r_1^2 = r_2^2 + r_3^2 \), then \( \theta_2 \) is equal to:

If a line along a chord of the circle \( 4x^{2}+4y^{2}+120x+675=0 \), passes through the point \( (-30,0) \), and is tangent to the parabola \( y^{2}=30x \), then the length of this chord is:

Let the lines \( y + 2x = \sqrt{11} + 7\sqrt{7} \) and \( 2y + x = 2\sqrt{11} + 6\sqrt{7} \) be normal to a circle \( C : (x - h)^2 + (y - k)^2 = r^2 \). If the line \( \sqrt{11}y - 3x = \frac{5\sqrt{77}}{3} + 11 \) is tangent to the circle \( C \), then the value of \( (5h - 8k)^2 + 5r^2 \) is equal to ________.

Let and \( \det(A-aI)=0 \), where \( a \) is a real number. If the largest possible value of \( a \) is \( p \), then the circle \( (x-p)^2+(y-2p)^2=320 \), intersects the co-ordinate axes at:

Let \( AB \) be a chord of length \( 12 \) of the circle \( (x-2)^{2}+(y+1)^{2}=\dfrac{169}{4} \). If tangents drawn to the circle at point \( A \) and \( B \) intersect at the point \( P \), then five times the distance of point \( P \) from chord \( AB \) is equal to ________.

Let \( ABCD \) and \( AEFG \) be squares of side \( 4 \) and \( 2 \) units, respectively. The point \( E \) is on the line segment \( AB \) and the point \( F \) is on the diagonal \( AC \). Then the radius \( r \) of the circle passing through the point \( F \) and touching the line segments \( BC \) and \( CD \) satisfies:

Consider two circles \( C_{1}: x^{2}+y^{2}=25 \) and \( C_{2}:(x-\alpha)^{2}+y^{2}=16 \), where \( \alpha \in (5,9) \). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of \( C_{1} \) and \( C_{2} \) be \( \sin^{-1}\!\left(\dfrac{\sqrt{63}}{8}\right) \). If the length of common chord of \( C_{1} \) and \( C_{2} \) is \( \beta \), then the value of \( (\alpha \beta^{2}) \) equals ______.

Choose the incorrect statement about the two circles whose equations are given below: \( x^{2} + y^{2} - 10x - 10y + 41 = 0 \) and \( x^{2} + y^{2} - 16x - 10y + 80 = 0 \)

Let a circle \( C : (x - h)^{2} + (y - k)^{2} = r^{2}, \ k > 0, \) touch the \( x \)-axis at \( (1, 0) \). If the line \( x + y = 0 \) intersects the circle \( C \) at \( P \) and \( Q \) such that the length of the chord \( PQ \) is \( 2 \), then the value of \( h + k + r \) is equal to ___.

Let a circle \( C \) of radius \( 5 \) lie below the \( x \)-axis. The line \( L_{1}: 4x+3y+2=0 \) passes through the centre \( P \) of the circle \( C \) and intersects line \( L_{2}: 3x-4y-11=0 \) at \( Q \). The line \( L_{2} \) touches \( C \) at the point \( Q \). Then the distance of \( P \) from the line \( 5x-12y+51=0 \) is ______.

If the variable line \( 3x + 4y = \alpha \) lies between the two circles \( (x - 1)^2 + (y - 1)^2 = 1 \) and \( (x - 9)^2 + (y - 1)^2 = 4 \), without intercepting a chord on either circle, then the sum of all the integral values of \( \alpha \) is __________ .

Let the line \( x-y=4 \) intersect the circle \( C:(x-4)^{2}+(y+3)^{2}=9 \) at the points \( Q \) and \( R \). If \( P(\alpha,\beta) \) is a point on \( C \) such that \( PQ=PR \), then \( (6\alpha+8\beta)^{2} \) is equal to __________.

Two tangents are drawn from the point \( P(-1,1) \) to the circle \( x^2 + y^2 - 2x - 6y + 6 = 0 \). If these tangents touch the circle at points \( A \) and \( B \), and if \( D \) is a point on the circle such that length of the segments \( AB \) and \( AD \) are equal, then the area of triangle \( ABD \) is equal to:

Let a circle \( C \) of radius \( 1 \) and closer to the origin be such that the lines passing through the point \( (3, 2) \) and parallel to the coordinate axes touch it. Then the shortest distance of the circle \( C \) from the point \( (5, 5) \) is:

The number of common tangents, to the circles \( x^2 + y^2 - 18x - 15y + 131 = 0 \) and \( x^2 + y^2 - 6x - 6y - 7 = 0 \), is:

Consider the circle \( C:x^2+y^2-6x-8y-11=0 \). Let a variable chord \( AB \) of the circle \( C \) subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from the origin on the chord \( AB \) is the circle \( x^2+y^2-\alpha x-\beta y-\gamma=0 \), then \( \alpha+\beta+2\gamma \) is equal to \( \_\_\_\_\_ \).

Let the set of all values of \( r \), for which the circles \( (x + 1)^2 + (y + 4)^2 = r^2 \) and \( x^2 + y^2 - 4x - 2y - 4 = 0 \) intersect at two distinct points be the interval \( (\alpha, \beta) \). Then \( \alpha \beta \) is equal to:

Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \dfrac{\text{length of arc }AB}{\text{length of arc }BC}=\dfrac{1}{5} \), and \( \overrightarrow{OC}=\alpha\,\overrightarrow{OA}+\beta\,\overrightarrow{OB} \), then \( a+\sqrt{2}(\sqrt{3}-1)\beta \) is equal to:

Let a circle \( C \) have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of \( C \) on the line \( x+y=1 \) is \( \sqrt{14} \), then the square of the radius of \( C \) is ________.

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point \( (2,-3) \) from the line \( 3x+4y=5 \), is given by:

Consider a circle \( C_{1}: x^{2}+y^{2}-4x-2y=\alpha-5 \). Let its mirror image in the line \( y=2x+1 \) be another circle \( C_{2}:5x^{2}+5y^{2}-10fx-10gy+36=0 \). Let \( r \) be the radius of \( C_{2} \). Then \( \alpha+r \) is equal to ________.

Let the locus of the midpoints of the chords of the circle \( x^2 + (y-1)^2 = 1 \) drawn from the origin intersect the line \( x + y = 1 \) at \( P \) and \( Q \). Then, the length of \( PQ \) is:

A circle \( C \) of radius \( 2 \) lies in the second quadrant and touches both the coordinate axes. Let \( r \) be the radius of a circle that has centre at the point \( (2,5) \) and intersects the circle \( C \) at exactly two points. If the set of all possible values of \( r \) is the interval \( (\alpha,\beta) \), then \( 3\beta-2\alpha \) is equal to:

Let the locus of the centre \( (\alpha, \beta) \), \( \beta > 0 \), of the circle which touches the circle \( x^2 + (y-1)^2 = 1 \) externally and also touches the \( x \)-axis be \( L \). Then the area bounded by \( L \) and the line \( y = 4 \) is:

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JEE Main 2025 2nd April Morning Shift

The absolute difference between the squares of the radii of the two circles passing through the point \( (-9,\,4) \) and touching the lines \( x+y=3 \) and \( x-y=3 \), is equal to __________.