Question

The locus of the moving point $P$ such that $2PA=3PB$, where $A$ is $(0,0)$ and $B$ is $(4,-3)$, is

• A. $5x^2+5y^2+72x+54y+225=0$
• B. $5x^2+5y^2-72x-54y+225=0$
• C. $5x^2+5y^2-72x+54y+225=0$
• D. $5x^2+5y^2-72x+54y-225=0$

Answer 1

The correct option is C $5x^2+5y^2-72x+54y+225=0$

Let the coordinates of the point $P$ be $(\alpha ,\beta )$.
Then, we have
$2PA=3PB\Rightarrow 4P{A}^2=9P{B}^2$
$\Rightarrow 4\left[\right(\alpha -0{)}^2+(\beta -0{)}^2]=9\left[\right(\alpha -4{)}^2+(\beta +3{)}^2]$
$\Rightarrow 5{\alpha }^2+5{\beta }^2-72\alpha +54\beta +225=0$
$\therefore$ The locus of $P$ is $5x^2+5y^2-72x+54y+225=0$