Polar of a Point

Polar of (x₁, y₁) with respect to the ellipse. Coincides with the chord of contact when the point is external, and with the tangent when the point is on the ellipse. La Hire's theorem holds.

The polar of point $P(x_1, y_1)$ with respect to $x^2/a^2 + y^2/b^2 = 1$ is defined as the locus of points of intersection of tangents at pairs of points $A$, $B$ on the ellipse such that chord $AB$ passes through $P$.

Derivation: Let $A(x_2, y_2)$ and $B(x_3, y_3)$ be on the ellipse with chord $AB$ passing through $P(x_1, y_1)$.

The tangent at $A$ is $xx_2/a^2 + yy_2/b^2 = 1$. Its pole is $A(x_2, y_2)$. The tangent at $B$ similarly has pole $B$.

The intersection $Q(h, k)$ of the tangents at $A$ and $B$ lies on the chord of contact from $Q$:

$$\frac{xh}{a^2} + \frac{yk}{b^2} = 1$$

This chord passes through $P(x_1, y_1)$:

$$\frac{x_1h}{a^2} + \frac{y_1k}{b^2} = 1$$

The locus of $Q(h, k)$ is therefore $\dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1$ — the polar.

La Hire's theorem: If $Q$ lies on the polar of $P$, then $P$ lies on the polar of $Q$.

Pole of a line $lx + my + n = 0$: Compare with $xx_1/a^2 + yy_1/b^2 = 1$ (written as $xx_1/a^2 + yy_1/b^2 - 1 = 0$):

$$\frac{x_1/a^2}{l} = \frac{y_1/b^2}{m} = \frac{-1}{n} \implies x_1 = -\frac{a^2l}{n}, \quad y_1 = -\frac{b^2m}{n}$$