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

Each of the two orthogonal circles $C_1$ and $C_2$ passes through both the points $(2,0)$ and $(-2,0)$. If $y=mx+c$ is a common tangent to these circles,then

The chord of contact of the tangents drawn from a point on the circle $x^2 + y^2 = a^2$ to the circle $x^2 + y^2 = b^2$ touches the circle $x^2 + y^2 = c^2$. Then $a, b, c$ are in:

The minimum distance between the curves $x^2 + y^2 + 4x + 16y + 66 = 0$ and $y^2 = 8x$ 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:

If two circles of the same radius $a$ and centers at $(2, 3)$ and $(5, 6)$ are orthogonal,find the value of $a$.