Equation of the tangent to the circle ${x^2} + {y^2} = {a^2}$ which is perpendicular to the straight line $y = mx + c$  is

Tangents are drawn from the point $(4, 3)$ to the circle ${x^2} + {y^2} = 9$. The area of the triangle formed by them and the line joining their points of contact is

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 line $ax + by + c = 0$ is a normal to the circle ${x^2} + {y^2} = {r^2}$. The portion of the line $ax + by + c = 0$ intercepted by this circle is of length

Line $y = x + a\sqrt 2 $ is a tangent to the circle ${x^2} + {y^2} = {a^2}$ at

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