If $P \left(2,-7\right)$ is a given point and Q is a point on $2x^2 + 9y^2 = 18$, then find the equation of the locus of the midpoint of PQ.
Let $Q \left(a, b\right)$ be a point on $2x^2 + 9y^2 = 18$
Then, $2a^2 + 9b^2 = 18$ $\;\;\; \cdots \; (1)$
Coordinates of midpoint of $P \left(2,-7\right)$ and $Q \left(a,b\right)$ are $\left(\dfrac{2 + a}{2}, \dfrac{b - 7}{2}\right)$
Let $Z \left(h,k\right)$ be a point on the required locus.
Then, $h = \dfrac{a + 2}{2}$, $k = \dfrac{b - 7}{2}$
i.e. $a = 2 h - 2$ $\;\;\; \cdots \; (2a)$ and $b = 2 k + 7$ $\;\;\; \cdots \; (2b)$
Substituting the values of $a$ and $b$ from equations $(2a)$ and $(2b)$ in equation $(1)$ gives
$2 \left(2h - 2\right)^2 + 9 \left(2 k + 7\right)^2 = 18$
i.e. $2 \left(4 h^2 - 8h + 4\right) + 9 \left(4 k^2 + 28 k + 49\right) = 18$
i.e. $8 h^2 - 16 h + 8 + 36 k^2 + 252 k + 441 = 18$
i.e. $8 h^2 + 36 k^2 - 16 h + 252 k + 431 = 0$
$\therefore$ $\;$ The equation of the required locus is: $\hspace{1em}$ $8x^2 + 36y^2 - 16 x + 252 y + 431 = 0$