Question

A point moves so that the ratio of its distances from 3x + 4y + 8 = 0 and 4x + 3y – 6 = 0 is 2. Find the equation of its locus.

Answer 1

A point moves so that the ratio of its distances from $3x + 4y + 8 = 0$ and $4x + 3y – 6 = 0$ is $2$. Let us find the equation of its locus. Let a point $(x,y)$ belongs to the locus. Then

$\frac{|3x + 4y + 8|}{\sqrt{3^2+4^2}}: \frac{|4x + 3y – 6|}{\sqrt{4^2+3^2}}=2.$

It follows that

$|3x + 4y + 8|=2|4x + 3y – 6|$

which is equivalent to

$|3x + 4y + 8|^2=4|4x + 3y – 6|^2.$

Then

$9x^2+16y^2+64+24xy+48x+64y=4(16x^2+9y^2+36+24xy-48x-36y).$

We conclude that

$55x^2+20y^2+72xy-144x-80y+80=0$

is the equation po its locus.