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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 \( C \)?

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

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 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:

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:

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:

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:

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:

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 \( \_\_\_\_\_\_ \).

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 \( ABCD \) 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 ______.

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:

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:

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JEE Main 2024 5th April Evening Shift

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 \( \_\_\_\_\_\_ \).