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

$$\begin{gathered} \therefore \frac{PA}{PB}=\frac{5}{4} \\ \Rightarrow \frac{\sqrt{{\left(h-2\right)}^2+{\left(k-0\right)}^2}}{\sqrt{{\left(h-1\right)}^2+{\left(k-3\right)}^2}}=\frac{5}{4} \end{gathered}$$

Squaring both sides, we get:

$$\begin{gathered} 16\left(h^2-4h+4+k^2\right)=25\left(h^2-2h+1+k^2-6k+9\right) \\ \Rightarrow 9h^2+9k^2+64h-50h-150k-64+250=0 \\ \Rightarrow 9h^2+9k^2+14h-150k+186=0 \end{gathered}$$

Hence, the locus of (h, k) is $9x^2+9y^2+14x-150y+186=0$.