Answer
$(4,0),(-4,0),(0,4)$
Work Step by Step
$x^{2}+y^{2} = 16$ Equation $(1)$
$y=-\frac{1}{4}x^{2}+4$ Equation $(2)$
From Equation $(2)$
$y=-\frac{1}{4}x^{2}+4$
Taking LCD,
$y=\frac{-x^{2}+16}{4}$
$4y=-x^{2}+16$
$x^{2}=16-4y$ Equation $(3)$
Substituting $x^{2}=16-4y$ in Equation $(1)$
$x^{2}+y^{2} = 16$
$16-4y+y^{2} = 16$
$16-4y+y^{2} -16=0$
$y^{2} -4y= 0$
$y(y -4)= 0$
$y=0$ or $y=4$
Substituting $y$ values in Equation $(3)$ to get $x$
Let $y=0$
$x^{2}=16-4y$
$x^{2}=16-4(0)$
$x^{2}=16$
$x=±4$
Let $y=4$
$x^{2}=16-4y$
$x^{2}=16-4(4)$
$x^{2}=0$
$x=0$
$(4,0),(-4,0),(0,4)$ satisfy the given equations.
The solutions are $(4,0),(-4,0),(0,4)$