Question

Let $A\equiv (-3,0)$ and $B\equiv (3,0)$ be two fixed points and $P$ moves on a plane such that $PA=n\times PB(n>0)$. If $n\ne 1$, then locus of a point $P$ is

• A. a straight line
• B. a circle
• C. a parabola
• D. an ellipse

Answer 1

The correct option is B a circle

Given, $A=(-3,0),B=(3,0)$ and $PA=nPB$

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

$PA=(h+3,k)$

$PB=(h-3,k)$

So, $PA=nPB$

$\Rightarrow$${(h+3)}^2+k^2=n^2[{(h-3)}^2+k^2]$

$\therefore$ locus of $P(h,k)$ is

$\Rightarrow$ $x^2+6x+9+y^2=n^2[x^2-6x+9+y^2]$

$\Rightarrow$$x^2(1-n^2)+y^2(1-n^2)+6x(1+n^2)+9(1-n^2)=0$

$\Rightarrow$$x^2+y^2+6\frac{(1+n^2)}{1-n^2}x+9=0(\because n\ne 1)$

$\therefore$ locus is a circle