Answer
$(6,0),(-6,0),(0,-6)$
Work Step by Step
$x^{2}+y^{2} = 36$ Equation $(1)$
$y=\frac{1}{6}x^{2}-6$ Equation $(2)$
From Equation $(2)$
$y=\frac{1}{6}x^{2}-6$
Taking LCD,
$y=\frac{x^{2}-36}{6}$
$6y=x^{2}-36$
$6y+36=x^{2}$ Equation $(3)$
Substituting $x^{2}=6y+36$ in Equation $(1)$
$x^{2}+y^{2} = 36$
$6y+36+y^{2} = 36$
$y^{2}+6y+36- 36=0$
$y^{2}+6y=0$
$y(y+6)=0$
$y=0$ or $y=-6$
Substituting $y$ values in Equation $(3)$ to get $x$
Let $y=0$
$x^{2}=6y+36$
$x^{2}=6(0)+36$
$x^{2}=36$
$x=±6$
Let $y=-6$
$x^{2}=6y+36$
$x^{2}=6(-6)+36$
$x^{2}=-36+36$
$x^{2}=0$
$x=0$
$(6,0),(-6,0),(0,-6)$ satisfy the given equations. The solutions are
$(6,0),(-6,0),(0,-6)$
