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

At which point on $y$-axis the line $x = 0$ is a tangent to circle ${x^2} + {y^2} - 2x - 6y + 9 = 0$

Points $P (-3,2), Q (9,10)$ and $R (\alpha, 4)$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - ky =1$, then $k$ is equal to $.........$.

Length of the tangent drawn from any point on the circle ${x^2} + {y^2} + 2gx + 2fy + {c_1} = 0$ to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ is

The equation of the tangent to the circle ${x^2} + {y^2} = {r^2}$ at $(a,b)$ is $ax + by - \lambda = 0$, where $\lambda $ is