The locus of the centers of the circles such that the point $(2, 3)$ is the midpoint of the chord $5x + 2y = 16$ 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 equation of the common tangent to the parabola $y^2=8x$ and the circle $x^2+y^2=2$ is $ax+by+2=0$. If $-\frac{a}{b} > 0$,then $3a^2+2b+1=$

Let $C_i \equiv x^2 + y^2 = i^2$ for $i = 1, 2, 3$ be three circles. There are $4i$ points on the circumference of each circle $C_i$. If no three of all the points on the three circles are collinear,then the number of triangles that can be formed using these points whose circumcentre does not lie on the origin is:

The common chord of two intersecting circles $c_1$ and $c_2$ subtends angles of $90^\circ$ and $60^\circ$ at their respective centres. If the distance between their centres is $\sqrt{3} + 1$,then the radii of $c_1$ and $c_2$ are:

The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $X$-axis is