Square of the length of the tangent drawn from the point $(\alpha ,\beta )$ to the circle $a{x^2} + a{y^2} = {r^2}$ is

If the centre of a circle is $(2, 3)$ and a tangent is $x + y = 1$, then the equation of this circle 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

The equation of pair of tangents to the circle ${x^2} + {y^2} – 2x + 4y + 3 = 0$ from $(6, – 5)$, is

Two tangents drawn from the origin to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ will be perpendicular to each other, if