Question

Find the equation to the locus of a point which moves so that its distance from the axis of $x$ is three times its distance from the axis of $y$.

Answer 1

Let the point be $P(h,k)$

$PA$ be the distance from $x$ axis and $PB$ be the distance from $y$ axis

Equation of $x$ axis is $y=0$

$PA=\frac{0\left(h\right)+1\left(k\right)+0}{\sqrt{0^2+1^2}}=k$

Equation of $y$ axis is $x=0$

$PB=\frac{1\left(h\right)+0\left(k\right)+0}{\sqrt{1^2+0^2}}=h$

Given $PA=3PB$

$\Rightarrow k=3h$

Replacing $h$ by $x$ and $k$ by $y$

$y=3x$