Question

Three distinct points are given in the 2-dimensional coordinate plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $\frac{1}{3}$. Then the locus of point $A$

• A. $x^2-y^2-\frac{5}{2}x+1=0$
  
• B. $x^2+y^2-\frac{5}{2}x+1=0$
  
• C. $x^2+y^2+\frac{5}{2}x+1=0$
  
• D. None of these

Answer 1

The correct option is B $x^2+y^2-\frac{5}{2}x+1=0$

Let the point $A$, $P$ and $Q$ are $(h,k)$, $(1,0)$ and $(-1,0)$ respectively.
$\frac{AP}{AQ}=\frac{1}{3}$
$\Rightarrow 3AP=AQ$
$\Rightarrow 9A{P}^2=A{Q}^2$
$\Rightarrow 9\left[\right(h-1{)}^2+k^2]=(h+1{)}^2+k^2$
$\Rightarrow h^2+k^2-\frac{5}{2}h+1=0$
Locus of $A(h,k)$ is
$x^2+y^2-\frac{5}{2}x+1=0$