If the line $lx + my + n = 0$ be a tangent to the circle ${(x - h)^2} + {(y - k)^2} = {a^2},$ then

The equation of the tangent at the point $\left( {\frac{{a{b^2}}}{{{a^2} + {b^2}}},\frac{{{a^2}b}}{{{a^2} + {b^2}}}} \right)$ of the circle ${x^2} + {y^2} = \frac{{{a^2}{b^2}}}{{{a^2} + {b^2}}} $ is

The length of tangent from the point $(5, 1)$ to the circle ${x^2} + {y^2} + 6x - 4y - 3 = 0$, is

Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6 x+4 y+8=0$ at the point $P$ and $Q$ on it. If the circumcircle of the triangle OPQ passes through the point $\left(\alpha, \frac{1}{2}\right)$, then a value of $\alpha$ is

If the lengths of the chords intercepted by the circle ${x^2} + {y^2} + 2gx + 2fy = 0$ from the co-ordinate axes be $10$ and $24$ respectively, then the radius of the circle is..

The co-ordinates of the point from where the tangents are drawn to the circles ${x^2} + {y^2} = 1$, ${x^2} + {y^2} + 8x + 15 = 0$ and ${x^2} + {y^2} + 10y + 24 = 0$ are of same length, are