Polar of a Point

Polar of (x₁, y₁) with respect to x²/a² − y²/b² = 1. Same form as the chord of contact and the tangent at a point. La Hire's theorem holds.

The polar of $P(x_1, y_1)$ with respect to $x^2/a^2 - y^2/b^2 = 1$ is:

$$\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$$

Derivation: Let $A(x_2, y_2)$ and $B(x_3, y_3)$ be on the hyperbola with chord $AB$ through $P$. The tangents at $A$ and $B$ meet at $Q(h, k)$.

The chord of contact from $Q$ is $xh/a^2 - yk/b^2 = 1$. Since 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 $xx_1/a^2 - yy_1/b^2 = 1$.

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

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

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

The minus sign in the pole's $y$-coordinate distinguishes the hyperbola from the ellipse.