12

12. [Maximum mark: 22]

The curve $C$ has equation $4x^2 + y^2 - 24x + 4y + 20 = 0$. The following diagram shows $C$ with a maximum point at A. (Diagram shows an ellipse centered at (3, -2)).

(a) Use implicit differentiation to show that $\displaystyle\frac{dy}{dx} = \frac{4(3 - x)}{y + 2}$. [4]

(b) Hence, determine the domain of $C$. Give your answer in the form $3 - \sqrt{a} \le x \le 3 + \sqrt{a}$, where $a \in \mathbb{Z}^+$. [4]

(c) Find $(x_A, y_A)$, the coordinates of A. [3]

A line $y = mx$ is a tangent to $C$, where $m \in \mathbb{Z}$.

(d) Find the possible values of $m$. [4]

The line $y = -4x$ touches $C$ at point B.

(e) Find $y_B$, the $y$‑coordinate of B. [3]

The region bounded by the curve $C$, the $y$‑axis and the lines $y = y_A$ and $y = y_B$, is rotated 360° about the $y$‑axis to form a solid of revolution.

(f) Find the volume of the solid formed. [4]