The number of circles touching the line $y - x = 0$ and the $y$-axis is
Zero
One
Two
Infinite
If a circle passes through the point $(1, 2)$ and cuts the circle ${x^2} + {y^2} = 4$ orthogonally, then the equation of the locus of its centre is
The condition that the circle ${(x - 3)^2} + {(y - 4)^2} = {r^2}$ lies entirely within the circle ${x^2} + {y^2} = {R^2},$ is
If the circles ${x^2} + {y^2} + 2x + 2ky + 6 = 0$ and ${x^2} + {y^2} + 2ky + k = 0$ intersect orthogonally, then $k$ is
Let $S = 0$ is the locus of centre of a variable circle which intersect the circle $x^2 + y^2 -4x -6y = 0$ orthogonally at $(4, 6)$ . If $P$ is a variable point of $S = 0$ , then least value of $OP$ is (where $O$ is origin)
If the equation of the common tangent at the point $(1, -1)$ to the two circles, each of radius $13$, is $12x + 5y -7 = 0$, then the centre of the two circles are