Question

Find locus of a point so that its distance from the point (3, 0) is three times its distance from (0, 2).

Answer 1

Let the point be $P(h,k)$

Given points be $A(3,0)$ and $B(0,2)$

$PA=\sqrt{{(h-3)}^2+{(k-0)}^2}PB=\sqrt{{(h-0)}^2+{(k-2)}^2}$

Given $PA=3PB$

$\sqrt{{(h-3)}^2+{(k-0)}^2}=3\sqrt{{(h-0)}^2+{(k-2)}^2}{(h-3)}^2+{(k-0)}^2=9({(h-0)}^2+{(k-2)}^2)h^2+9-6h+k^2=9(h^2+k^2+4-4k)9h^2+9k^2+36-36k=h^2+9-6h+k^{2}8h^2+8k^2+6h-36k+27=0$

Replacing $h$ by $x$ and $k$ by $y$

$8x^2+8y^2+6x-36y+27=0$

is the required locus.