##
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 (*x*_{0},*y*_{0}) 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 (*x*_{0},*y*_{0}) 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 (*x*_{0},*y*_{0}) such that
,
we
will get

*a*^{2} = *b*^{2}

which clearly implies that the ellipse is a circle.

**
**

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