The equation of the normal at the point $(4,-1)$ of the circle $x^2+y^2-40 x+10 y=153$ is

The angle between the tangents drawn from the origin to the circle $(x -7)^2 + (y + 1)^2 = 25$ is :-

If the line $y = \sqrt 3 x + k$ touches the circle ${x^2} + {y^2} = 16$, then $k =$

From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the middle points of these chords is

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_1$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is

If the line $3x + 4y - 1 = 0$ touches the circle ${(x - 1)^2} + {(y - 2)^2} = {r^2}$, then the value of $r$ will be