Intersection of Tangents at Two Points

The tangents to y² = 4ax at parameters t₁ and t₂ intersect at (at₁t₂, a(t₁+t₂)). For a focal chord (t₁t₂ = −1), the tangents meet on the directrix.

Tangent at $(at_1^2, 2at_1)$: $\quad t_1 y = x + at_1^2 \quad \cdots (1)$

Tangent at $(at_2^2, 2at_2)$: $\quad t_2 y = x + at_2^2 \quad \cdots (2)$

Subtracting $(1)$ from $(2)$:

$$(t_2 - t_1)y = a(t_2^2 - t_1^2) = a(t_2-t_1)(t_2+t_1)$$

Since $t_1 \neq t_2$, divide by $(t_2 - t_1)$:

$$y = a(t_1 + t_2)$$

Substituting into $(1)$:

$$t_1 \cdot a(t_1+t_2) = x + at_1^2$$ $$x = a t_1(t_1+t_2) - at_1^2 = at_1 t_2$$

Therefore the tangents meet at:

$$\bigl(at_1t_2,\; a(t_1+t_2)\bigr)$$

Key special case — focal chord ($t_1 t_2 = -1$):

$$x\text{-coordinate} = at_1t_2 = -a$$

The tangents at the endpoints of any focal chord meet on the directrix $x = -a$. Furthermore, they meet at right angles (the angle between the tangents is 90°), confirming the directrix is the locus of intersection of perpendicular tangents — the director circle of the parabola degenerates to the directrix.