Chord of Contact

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

Let $P(x_1, y_1)$ be an external point and let $A(at_1^2, 2at_1)$ and $B(at_2^2, 2at_2)$ be the two points of tangency.

Tangent at $A$: $t_1 y = x + at_1^2$. Since it passes through $P(x_1, y_1)$:

$$t_1 y_1 = x_1 + at_1^2 \quad \cdots (1)$$

Tangent at $B$: $t_2 y = x + at_2^2$. Since it passes through $P(x_1, y_1)$:

$$t_2 y_1 = x_1 + at_2^2 \quad \cdots (2)$$

Both equations say that $(at^2, 2at)$ lies on the line:

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

(Verify: substitute $x = at^2$, $y = 2at$ — gives $2at \cdot y_1 = 2a(at^2 + x_1)$, i.e., $ty_1 = x_1 + at^2$, which matches equations (1) and (2).)

Therefore the chord of contact $AB$ has equation:

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

This is identical in form to the tangent at $(x_1, y_1)$. If $(x_1, y_1)$ is on the parabola, it gives the tangent; if external, the chord of contact. Context determines the interpretation.