Polar of a Point

Polar of the point (x₁, y₁) with respect to x²+y²=a². If the point lies on the circle, the polar is the tangent at that point. If outside, the polar is the chord of contact.

Let the circle be $x^2 + y^2 = a^2$ and the point be $P(x_1, y_1)$ (anywhere in the plane — inside, on, or outside).

Draw any chord $AB$ of the circle passing through $P$. Let tangents at $A$ and $B$ meet at $Q$. The locus of $Q$ as $AB$ rotates about $P$ is a straight line called the polar of $P$.

Derivation: Let $Q(h, k)$ be the intersection of tangents at $A$ and $B$. The chord of contact from $Q$ to $x^2+y^2=a^2$ is:

$$hx + ky = a^2$$

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

$$hx_1 + ky_1 = a^2$$

This means $Q(h,k)$ lies on the line $xx_1 + yy_1 = a^2$. The locus of $Q$ is therefore:

$$xx_1 + yy_1 = a^2$$

This is the polar of $P(x_1, y_1)$ with respect to $x^2 + y^2 = a^2$.

Three cases unified:

• $P$ outside: polar is the chord of contact (the line joining the two tangent contact points)
• $P$ on the circle: polar is the tangent at $P$ (the chord degenerates)
• $P$ inside: polar is a real line entirely outside the circle

La Hire's theorem: If $Q$ lies on the polar of $P$, then $P$ lies on the polar of $Q$. Pole and polar are a reciprocal pair.