Chord of Contact

Equation of the chord joining the two points of tangency when tangents are drawn from external point (x₁, y₁) to circle S = 0. Same form as the tangent at a point — context distinguishes the two.

Let $P(x_1, y_1)$ be an external point and $A(x_2, y_2)$, $B(x_3, y_3)$ the two points of tangency on circle $S \equiv x^2 + y^2 + 2gx + 2fy + c = 0$.

The tangent at $A(x_2, y_2)$ is:

$$xx_2 + yy_2 + g(x + x_2) + f(y + y_2) + c = 0$$

Since this tangent passes through $P(x_1, y_1)$:

$$x_1 x_2 + y_1 y_2 + g(x_1 + x_2) + f(y_1 + y_2) + c = 0 \quad \cdots (1)$$

Similarly, the tangent at $B(x_3, y_3)$ passes through $P$:

$$x_1 x_3 + y_1 y_3 + g(x_1 + x_3) + f(y_1 + y_3) + c = 0 \quad \cdots (2)$$

Both equations (1) and (2) say that $(x_2, y_2)$ and $(x_3, y_3)$ satisfy the single linear equation:

$$xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0$$

This is the chord of contact — the unique line passing through both contact points $A$ and $B$.

Observation: The chord of contact has exactly the same algebraic form as the tangent at a point. The distinction is contextual: if $(x_1, y_1)$ lies on the circle, the equation is a tangent; if $(x_1, y_1)$ is external, the same expression gives the chord of contact.