If the line $y = mx + c$be a tangent to the circle ${x^2} + {y^2} = {a^2}$, then the point of contact is

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B,$ then the locus of the foot of perpendicular from $O$ on $AB$ 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 $2 x ^2+2 y ^2-(1+ a ) x -(1- a ) y =0$ is equal to:

The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is

Point $M$ moved along the circle $(x - 4)^2 + (y - 8)^2 = 20 $. Then it broke away from it and moving along a tangent to the circle, cuts the $x-$ axis at the point $(- 2, 0)$ . The co-ordinates of the point on the circle at which the moving point broke away can be :

If $a > 2b > 0$ then the positive value of m for which $y = mx - b\sqrt {1 + {m^2}} $ is a common tangent to ${x^2} + {y^2} = {b^2}$ and ${(x - a)^2} + {y^2} = {b^2}$, is