Question

The locus of the point whose ratio of distance from the origin to its distance from $(-2,-3)$ is $5: 7$, is given by______

• A. $24\left(x^{2}+y^{2}\right)-100 x-150 y-325=0$
• B. $24\left(x^{2}+y^{2}\right)+100 x+150 y-325=0$
• C. $24\left(x^{2}+y^{2}\right)-100 x+150 y+325=0$
• D. $2 x^{2}+2 y^{2}=325$

Answer 1

Answer: (A)

Let the point $P(x, y)$, such that $O P: A P=5: 7$,
where $O$ is the origin and $A(-2,-3),$ so
$\frac{\sqrt{x^{2}+y^{2}}}{\sqrt{(x+2)^{2}+(y+3)^{2}}}=\frac{5}{7}$
$\Rightarrow 49\left(x^{2}+y^{2}\right)=25\left[x^{2}+y^{2}+4 x+6 y+13\right]$
$\Rightarrow 24\left(x^{2}+y^{2}\right)-100 x-150 y-375=0$