Tangent at Parametric Point t

Tangent to y² = 4ax at the point (at², 2at). Slope of this tangent is 1/t.

The point on $y^2 = 4ax$ with parameter $t$ is $(at^2, 2at)$. Substituting $x_1 = at^2$, $y_1 = 2at$ into the tangent formula $yy_1 = 2a(x + x_1)$:

$$y \cdot 2at = 2a(x + at^2)$$

Dividing both sides by $2a$:

$$ty = x + at^2$$

This is the tangent at parameter $t$.

Properties:

• Slope: rearranging, $y = \frac{1}{t}x + at$, so slope $= 1/t$
• $y$-intercept: $at$
• As $t \to 0$ (near vertex), the slope $\to \infty$ (tangent becomes vertical — the $y$-axis itself)
• As $t \to \infty$, slope $\to 0$ (tangent becomes nearly horizontal far out the parabola)

Two tangents are parallel iff their slopes are equal: $1/t_1 = 1/t_2 \Rightarrow t_1 = t_2$ (same point). So no two distinct points on a parabola have parallel tangents — a property unique to parabolas among conics.