Problem

The diagram shows the circle with centre $C(-4, 5)$ and radius $\sqrt{20}$ units. The circle intersects the $y$-axis at the points $A$ and $B$. The size of angle $ACB$ is $\theta$ radians.

(a) Find the equation of the tangent to the circle at the point $(-6, 9)$.

(b) Find the equation of the circle in the form $x^2 + y^2 + ax + by + c = 0$.

(c) Find the value of $\theta$ correct to 4 significant figures.

(d) Find the perimeter and area of the segment shaded in the diagram.