Polar of a Point

Polar of point (x₁, y₁) with respect to y² = 4ax. If (x₁, y₁) is on the parabola, the polar is the tangent there. If external, it is the chord of contact. La Hire's theorem holds.

The polar of point $P(x_1, y_1)$ with respect to $y^2 = 4ax$ is defined as the locus of the intersection of tangents at pairs of points $A$, $B$ on the parabola such that $P$ lies on chord $AB$.

Derivation: Let $A(at_1^2, 2at_1)$ and $B(at_2^2, 2at_2)$ be two points on the parabola such that chord $AB$ passes through $P(x_1, y_1)$.

The chord $AB$ has equation $y(t_1+t_2) = 2x + 2at_1t_2$. Passing through $P(x_1, y_1)$:

$$y_1(t_1+t_2) = 2x_1 + 2at_1t_2 \quad \cdots (*)$$

The tangents at $A$ and $B$ meet at $Q(at_1t_2,\, a(t_1+t_2))$ (from the intersection of tangents formula).

Let $Q = (h, k)$, so $h = at_1t_2$ and $k = a(t_1+t_2)$.

Substituting into $(*)$:

$$y_1 \cdot \frac{k}{a} = 2x_1 + 2h \implies ky_1 = 2a(x_1 + h)$$

The locus of $Q(h, k)$ is:

$$yy_1 = 2a(x + x_1)$$

This is the polar of $P(x_1, y_1)$.

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

Three interpretations of the same equation:

• $(x_1, y_1)$ on the parabola: tangent at that point
• $(x_1, y_1)$ external: chord of contact
• $(x_1, y_1)$ internal: polar line (no real tangents, but the polar is still a real line)