The area of triangle formed by the tangent, normal drawn at $(1,\sqrt 3 )$ to the circle ${x^2} + {y^2} = 4$ and positive $x$-axis, is

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

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

The equation of the tangents to the circle ${x^2} + {y^2} + 4x - 4y + 4 = 0$ which make equal intercepts on the positive coordinate axes is given by

A pair of tangents are drawn from the origin to the circle ${x^2} + {y^2} + 20(x + y) + 20 = 0$. The equation of the pair of tangents is

Equation of the tangent to the circle, at the point $(1 , -1)$ whose centre is the point of intersection of the straight lines $x - y = 1$ and $2x + y= 3$ is