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

Three concentric circles,of which the biggest is $x^2 + y^2 = 1$,have their radii in $A.P.$ If the line $y = x + 1$ cuts all the circles in real and distinct points,the interval in which the common difference $d$ of the $A.P.$ will lie is:

$A$ circle touches the parabola $y^2=4x$ at $(1,2)$ and also touches its directrix. The $y$-coordinate of the point of contact of the circle and the directrix is

The equations of the circles which touch the lines $3x - 4y + 1 = 0$ and $4x + 3y - 7 = 0$ and pass through the point $(2, 3)$ are:

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

Let $r$ be the radius of the circle,which touches the $x$-axis at point $(a, 0)$,where $a < 0$,and the parabola $y^2 = 9x$ at the point $(4, 6)$. Then $r$ is equal to . . . . . . .