$x = 7$ touches the circle ${x^2} + {y^2} - 4x - 6y - 12 = 0$, then the coordinates of the point of contact are

Let a circle $C$ of radius $5$ lie below the $x$-axis. The line $L_{1}=4 x+3 y-2$ passes through the centre $P$ of the circle $C$ and intersects the line $L _{2}: 3 x -4 y -11=0$ at $Q$. The line $L _{2}$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5 x-12 y+51=0$ is

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 equations of the tangents drawn from the point $(0, 1)$ to the circle ${x^2} + {y^2} - 2x + 4y = 0$ are

The slope of the tangent at the point $(h,h)$ of the circle ${x^2} + {y^2} = {a^2}$ is

The equation of the tangent to the circle ${x^2} + {y^2} = {a^2}$ which makes a triangle of area ${a^2}$ with the co-ordinate axes, is