Consider a circle $C_1: x^2+y^2-4 x-2 y=\alpha-5$.Let its mirror image in the line $y=2 x+1$ be another circle $C _2: 5 x ^2+5 y ^2-10 fx -10 gy +36=0$.Let $r$ be the radius of $C _2$. Then $\alpha+ r$ is equal to $......$.

If the curves, $x^{2}-6 x+y^{2}+8=0$ and $\mathrm{x}^{2}-8 \mathrm{y}+\mathrm{y}^{2}+16-\mathrm{k}=0,(\mathrm{k}>0)$ touch each other at a point, then the largest value of $\mathrm{k}$ 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 a circle passes through the point $(a , b) \&$ cuts the circle $x^2 + y^2= K^2$ orthogonally, then the equation of the locus of its centre is :

Let the latus ractum of the parabola $y ^{2}=4 x$ be the common chord to the circles $C _{1}$ and $C _{2}$ each of them having radius $2 \sqrt{5}$. Then, the distance between the centres of the circles $C _{1}$ and $C _{2}$ is

In the co-axial system of circle ${x^2} + {y^2} + 2gx + c = 0$, where $g$ is a parameter, if $c > 0$ then the circles are