## Implicit Differentiation - Exercise 3

Exercise 3. Show that if a normal line to each point on an ellipse passes through the center of an ellipse, then the ellipse is a circle.

Answer. An equation of the ellipse is given by where we assumed that (0,0) is the center. We may always do that. Let (x0,y0) be a point on the ellipse. The slope of the tangent line to the ellipse at this point will be obtained through implicit differentiation. Indeed, we have or equivalently So the slope of the tangent line at the point (x0,y0) is which gives the slope of the normal line as . Hence the equation of the normal line is or equivalently Assuming that (0,0) is on any normal line, we get If we choose a point (x0,y0) such that , we will get

a2 = b2

which clearly implies that the ellipse is a circle.

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