微積分應用題目，請問圈起來的怎麼解?

63.
x² + y² = r²
centre = (0, 0) radius = r

x² + y² = r²
(d/dx)(x² + y²) = (d/dx)r²
2x + 2y(dy/dx) = 0
2y(dy/dx) = -2x
dy/dx = -x/y

Consider an arbitrary point (a, b) on the circle.

Slope of the tangent line at (a, b) = -a/b
Slope of the normal line at (a, b) = -1/(-a/b) = b/a

Equation of the normal line at (a, b) :
y - b = (b/a)(x - a)
a(y - b) = b(x - a)
ay - ab = bx - ab
ay = bx
y = (b/a)x + 0

y-intercept of the normal line at (a, b) = 0
Hence, the normal line at (a, b) passes through the origin (0, 0).

As (a, b) is an arbitrary point on the circle, the normal line at any point on the circle passes through the origin.