Question

Find the equation to the locus of a point so that the sum of the squares of its distances from the axes is equal to $3$.

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 ${PA}^2+{PB}^2=3$

$k^2+h^2=3h^2+k^2=3$

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

$x^2+y^2=3$

is the required locus.