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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 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 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 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 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.

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 :

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 \( A \) 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 :

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 _____.

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 \( P(a_{1},b_{1}) \) 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 \( A \) 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:

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:

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:

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 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:

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 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 ________.

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JEE Main 2023 15th April Morning Shift

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: