The normal to the circle ${x^2} + {y^2} - 3x - 6y - 10 = 0$at the point $(-3, 4)$, is

If $\theta $ is the angle subtended at $P({x_1},{y_1})$ by the circle $S \equiv {x^2} + {y^2} + 2gx + 2fy + c = 0$, then

The equation of the normal to the circle ${x^2} + {y^2} - 2x = 0$ parallel to the line $x + 2y = 3$ is

If the length of the tangents drawn from the point $(1,2)$ to the circles ${x^2} + {y^2} + x + y - 4 = 0$ and $3{x^2} + 3{y^2} - x - y + k = 0$ be in the ratio $4 : 3$, then $k =$

The equations of the tangents drawn from the origin to the circle ${x^2} + {y^2} - 2rx - 2hy + {h^2} = 0$ are

If the tangents at the points $P$ and $Q$ on the circle $x ^2+ y ^2-2 x + y =5$ meet at the point $R \left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is