Implicit derivitave of a general ellipse

Question

Consider an ellipse centered at the point $(h,k)$. Find all points $P=(x,y)$ on the ellipse for which the tangent line at $P$ is perpendicular to the line through $P$ and $(h,k)$. I know the general equation of an ellipse is: $$a^2(y-k)^2+b^2(x-h)^2=a^2b^2 .$$ I need to find the derivative of it, and that the derivative of the tangent line is: $$(a(y-k)/(-b(x-h)),$$ but what do I do from there?

Answer 1

Use implicit differentiation. Differentiate the equation of the ellipse with respect to $x$ to get $$2a^2(y-k)y' + 2b^2(x-h) = 0.$$ Thus the slope of the tangent line at any point $(x,y)$ on the ellipse is $$y' = -\frac{b^2(x-h)}{a^2(y-k)}.$$

Edit: unless $y=k$ in which case the tangent line is vertical.