Answer
$(12,-5),(-12,5)$
Work Step by Step
$x^{2}+y^{2}=169$ Equation $(1)$
$5x+12y=0$ Equation $(2)$
From Equation $(2)$,
$5x+12y=0$
$12y=-5x$
$y=\frac{-5x}{12}$
Substitute $y=\frac{-5x}{12}$ in Equation $(1)$
$x^{2}+y^{2}=169$
$x^{2}+(\frac{-5x}{12})^{2}=169$
$x^{2}+(\frac{25x^{2}}{144})=169$
Taking LCD,
$(\frac{144x^{2}+25x^{2}}{144})=169$
$144x^{2}+25x^{2}=169 \times 144$
$169x^{2}=169 \times 144$
$x^{2}=144$
$x=±12$
$x=12$ or $x=-12$
Substitute $x$ values in Equation $(2)$ to get corresponding $y$ values.
Let $x=12$
$5x+12y=0$
$5(12)+12y=0$
$60+12y=0$
$12y=-60$
$y=-5$
Let $x=-12$
$5x+12y=0$
$5(-12)+12y=0$
$-60+12y=0$
$12y=60$
$y=5$
$(12,-5),(-12,5)$ satisfy the given equations.
