The gradient of the normal at the point $(-2, -3)$ on the circle $x^2 + y^2 + 2x + 4y + 3 = 0$ is

Find the points on the curve $x^{2}+y^{2}-2x-3=0$ at which the tangents are parallel to the $x$-axis.

$A$ circle $C_{1}$ passes through the origin $O$ and has a diameter of $4$ on the positive $x$-axis. The line $y = 2x$ intersects the circle $C_{1}$ at $O$ and $A$. Let $C_{2}$ be the circle with $OA$ as a diameter. If the tangent to $C_{2}$ at the point $A$ meets the $x$-axis at $P$ and the $y$-axis at $Q$,then the ratio $QA : AP$ is equal to:

If the tangent at a point $P$ on the circle $x^2 + y^2 + 6x + 6y = 2$ meets the line $5x - 2y + 6 = 0$ at a point $Q$ on the $y$-axis,then the length of $PQ$ is . . . . .

$A$ circle touches the $y$-axis at the point $(0,4)$ and passes through the point $(2,0)$. Which of the following lines is not a tangent to this circle?

The gradient of the tangent line at the point $(a \cos \alpha, a \sin \alpha)$ to the circle $x^2 + y^2 = a^2$ is