How do you find the equations for the normal line to x^2+y^2=25 through (4,3)?

$y = \frac{3}{4} x$
This is the equation of a circle with center in the origin and $R = 5$.
For the properties of the circle, the radius is normal to the tangent, so the normal line will pass through $\left(0 , 0\right)$ as well as through $\left(4 , 3\right)$ and its equation is given by:
$y = \frac{3}{4} x$