Question

Find the equation of the locus of a point which moves such that the ratio of its distances from (2, 0) and (1, 3) is 5 : 4.

 

Answer 1

Let A(2, 0) and B(1, 3) be the given points. Let P (h, k) be a point such that PA:PB = 5:4
\[\therefore \frac{PA}{PB} = \frac{5}{4}\]
\[ \Rightarrow \frac{\sqrt{\left( h - 2 \right)^2 + \left( k - 0 \right)^2}}{\sqrt{\left( h - 1 \right)^2 + \left( k - 3 \right)^2}} = \frac{5}{4}\]
Squaring both sides, we get:
\[16\left( h^2 - 4h + 4 + k^2 \right) = 25\left( h^2 - 2h + 1 + k^2 - 6k + 9 \right)\]
\[ \Rightarrow 9 h^2 + 9 k^2 + 64h - 50h - 150k - 64 + 250 = 0\]
\[ \Rightarrow 9 h^2 + 9 k^2 + 14h - 150k + 186 = 0\]
Hence, the locus of (h, k) is
\[9 x^2 + 9 y^2 + 14x - 150y + 186 = 0\]