Chord of Contact

Chord joining the two points of tangency when tangents are drawn from external point (x₁, y₁). Same form as the tangent at a point — context determines interpretation.

Let $P(x_1, y_1)$ be external to the ellipse $x^2/a^2 + y^2/b^2 = 1$, and let tangents from $P$ touch the ellipse at $A(x_2, y_2)$ and $B(x_3, y_3)$.

Tangent at $A$: $xx_2/a^2 + yy_2/b^2 = 1$. Since it passes through $P$:

$$\frac{x_1x_2}{a^2} + \frac{y_1y_2}{b^2} = 1 \quad \cdots (1)$$

Tangent at $B$: Similarly, $\dfrac{x_1x_3}{a^2} + \dfrac{y_1y_3}{b^2} = 1 \quad \cdots (2)$

Both equations say that $A$ and $B$ satisfy the linear equation in $(x, y)$:

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

This single line passes through both contact points — it is the chord of contact.

As with all conics, the chord of contact, tangent at a point, and polar of a point all share the same algebraic form (T = 0). The geometric context determines which one applies.