Polar curve of a conic with respect to a circle

Question

Given a circle of center $O$ and radius $r$, I know that the polar curve of a conic with respect to the circle is, in general, a conic. Now, in a work about Kepler's laws, I found that: if the center of the circle coincide with a focus of the conic, the polar curve is a circle. Someone know how we can prove this claim?

Answer 1

$19$th and early $20$th century textbooks for the win!

You can find a synthetic proof in Russel, Pure Geometry, pg 93. You may have to go to earlier parts of the book to get lemmas and definitions.

This is part of a chapter on the polar reciprocation of conics in both circles and conics. (Polar reciprocals are what we now call polar curves.)

A variant of the proof is in Pickford, Projective Geometry, $\S 187$.

There is a different proof at Casey, Analytical Geometry, $\S 125$ (pg 184). Again, it may require referring to earlier material.

Answer 2

I haven't read the books suggested in the other answer, but I found a fairly simple proof - apologies if it is the same given there.

Suppose we have an ellipse with focus $F$ and a circle $c$ (pink in figure below) centred at $F$. To construct the polar line of a point $P$ on the ellipse, with respect to $c$, we can construct point $P'$, the circular inverse of $P$ with respect to $c$: the polar is then the line perpendicular to $FP$ at $P'$. The polar of the ellipse is the envelope of all the polars of $P$, as $P$ varies on the ellipse.

To construct the point on the envelope lying on the polar line of $P$, we can consider another point $Q$ on the ellipse, near to $P$: as $Q\to P$, the intersection of the polar lines of $P$ and $Q$ tends to a point $M$ on the envelope.

To find $M$, it is equivalent (but easier) to consider point $Q$ on the tangent at $P$ to the ellipse. If $Q'$ is the inverse of $Q$, then points $FP'Q'$ lie on a circle, which is the inverse of the tangent, and $M$ is the other endpoint of the diameter through $F$ (because $\angle FPM=\angle FQM=90°$). This means that $M$ is the inverse of point $H$, which is the projection of $F$ on the tangent.

But it is easy to prove (see here for a proof) that the locus of the projections of a focus on the tangents to an ellipse is its auxiliary circle, that is the circle having the major axis of the ellipse as diameter (solid green in the figure). Hence the polar of the ellipse, which is the locus of $M$, is indeed a circle: the inverse of the auxiliary circle with respect to $c$ (dashed green in the figure).