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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 ________.

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:

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 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:

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

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:

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 \( r \) 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 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 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 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:

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:

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JEE MAIN 2025 4th April Morning Shift

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 ______.