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 

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