Question

Find the equation of the set of all points whose distances from (0, 4) are $\frac{2}{3}$ of their distances from the line y = 9.

Answer 1

We have
$$\begin{gathered} \mathrm{PQ}=\frac{2}{3}\mathrm{PL} \\ \Rightarrow \sqrt{{\left(x-0\right)}^2+{\left(y-4\right)}^2}=\frac{2}{3}\left(y-9\right) \\ \Rightarrow 3^2\left[x^2+{\left(y-4\right)}^2\right]=2^2{\left(y-9\right)}^2 \\ \Rightarrow 9x^2+9y^2-72y+144=4y^2-72y+324 \\ \Rightarrow 9x^2+5y^2=180 \\ \Rightarrow \frac{x^2}{20}+\frac{y^2}{36}=1 \end{gathered}$$