Consider circle $S$ : $x^2 + y^2 = 1$ and $P(0, -1)$ on it. $A$ ray of light gets reflected from tangent to $S$ at $P$ from the point with abscissa $-3$ and becomes tangent to the circle $S.$  Equation of reflected ray is

The line $x = y$ touches a circle at the point $(1, 1)$. If the circle also passes through the point $(1, -3)$, then its radius is

If the line $x = k$ touches the circle ${x^2} + {y^2} = 9$, then the value of $k$ is

A circle with centre $(2,3)$ and radius $4$ intersects the line $x + y =3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S(\alpha, \beta)$, then $4 \alpha-7 \beta$ is equal to $........$.

Tangents are drawn from $(4, 4) $ to the circle $x^2 + y^2 - 2x - 2y - 7 = 0$ to meet the circle at $A$ and $B$. The length of the chord $AB $ 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