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
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
Assuming that (0,0) is on any normal line, we get
If we choose a point (x0,y0) such that
a2 = b2
which clearly implies that the ellipse is a circle.
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