Find the locus!

Geometry Level pending

A and B are two fixed points . P is a moving point such that PA= nPB where n is a positive integer greater than one.

What is the locus of P ?

Hyperbola Circle Straight line Conic Parabola Ellipse

1 solution

Nashita Rahman
Oct 24, 2018

Let us take A(0,0) , B(a,0).

$\frac{PA^{2}}{PB^{2}}$ = $n^{2}$ (since it is given that PA=nPB)

Or, $\frac{x^{2}+ y^{2}}{(x-a)^{2}+y^{2}}$ = $n^{2}$ where P is (x,y)

Or, $x^{2} + y^{2}$ = $n^{2}$ [ $(x-a)^{2} + y^{2}$ ] is the locus of P

Or, ( $n^{2}$ -1) $x^{2}$ + ( $n^{2}$ -1) $y^{2}$ -2 $n^{2}$ ax + $n^{2}a^{2}$ = 0 which is a circle.

Hence, the locus of P is a circle.