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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 ________.

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

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 \( C \) 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:

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 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 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:

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 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 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:

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 :

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:

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:

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

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 ______.

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 ______.

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 ____________ .

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 \( AB \) 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 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 ______.

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 2022 29th June Evening Shift

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: