Question

Find the locus of a variable point p, if the distance of p from the x-axis is always three times its distance from the point (3, -3)?​

Answer 1

Let $(x,y)$ be the coordinates of point P.

The distance from the x-axis is exactly the y coordinates of point P.

The distance from the point (3,-3) is given by

$\sqrt{(x-3)^2+(x+3)^2}$

So, we want

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

We can square both sides to get

$y^2=9((x-3)^2+(y+3)^2)$

Expanding the squares, we have

$y^2=9(x^2-6x+9+y^2+6y+9) = 9(x^2+y^2-6x+6y+18)=9x^2+9y^2-54x+54y+162$

So, the final equation is

$y^2=9x^2+9y^2-54x+54y+162 \iff 9x^2+8y^2-54x+54y+162=0$