A circle with centre $(a, b)$ passes through the origin. The equation of the tangent to the circle at the origin is

The length of the tangent from the point $(4, 5)$ to the circle ${x^2} + {y^2} + 2x - 6y = 6$ is

If a line passing through origin touches the circle ${(x - 4)^2} + {(y + 5)^2} = 25$, then its slope should be

Let the tangent to the circle $C _{1}: x^{2}+y^{2}=2$ at the point $M (-1,1)$ intersect the circle $C _{2}$ : $( x -3)^{2}+(y-2)^{2}=5$, at two distinct points $A$ and $B$. If the tangents to $C _{2}$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $ANB$ is equal to

The equation of the circle having the lines $y^2 - 2y + 4x - 2xy = 0$ as its normals $\&$ passing through the point $(2 , 1)$ is :

Let $PQ$ and $RS$ be the tangent at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle, then $(PQ.RS)$ is equal to