The point at which the normal to the circle ${x^2} + {y^2} + 4x + 6y – 39 = 0$ at the point $(2, 3)$ will meet the circle again, is

Square of the length of the tangent drawn from the point $(\alpha ,\beta )$ to the circle $a{x^2} + a{y^2} = {r^2}$ is

If the line $lx + my = 1$ be a tangent to the circle ${x^2} + {y^2} = {a^2}$, then the locus of the point $(l, m)$ is

Consider a circle $(x-\alpha)^2+(y-\beta)^2=50$, where $\alpha, \beta>0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4 \sqrt{2}$ , then $(\alpha+\beta)^2$ is equal to…………….

The equation of the normal to the circle ${x^2} + {y^2} – 2x = 0$ parallel to the line $x + 2y = 3$ is