Question

Find the equation of the locus of a point which moves such that the ratio of its distances from $(2,0)and(1,3)$ is $5:4$.

Answer 1

Let the point be $P(x,y)$ and the mid points $A(2,0)$ and $B(1,3)$

then according to the distance formula

$PA=\sqrt{(x-2{)}^2+(y-0{)}^2}$

$PB=\sqrt{(x-1{)}^2+(y-3{)}^2}$

Given that

$\frac{PA}{PB}=\frac{5}{4}$

$\frac{P{A}^2}{P{B}^2}=\frac{25}{16}$

$\frac{(x-2{)}^2+y^2}{(x-1{)}^2+(y-3{)}^2}=\frac{25}{16}$

$\frac{x^2+4-4x+y^2}{x^2+1-2x+y^2+9-6y}=\frac{25}{16}$

$16x^2+64-64x+16y^2=25x^2+25-50x+25y^2+225-150y$

$25x^2-16x^2+25y^2-16y^2+64x-50x-150y+225-64+25=0$

$9x^2+9y^2+14x-150y+186=0$