Focal Distance

Distance from any point P(x₁, y₁) on y² = 4ax to the focus S(a, 0). Equals the distance from P to the directrix — the defining property of a parabola.

For $y^2 = 4ax$, focus $S(a, 0)$, directrix $x = -a$.

Let $P(x_1, y_1)$ be on the parabola (so $y_1^2 = 4ax_1$). The perpendicular from $P$ to the directrix has foot $M(-a, y_1)$.

By definition of a parabola:

$$PS = PM = x_1 - (-a) = x_1 + a$$

Therefore the focal distance is simply:

$$PS = x_1 + a$$

Consequences:

• Since $x_1 \geq 0$ for $y^2 = 4ax$, the minimum focal distance is $a$ (at the vertex). This minimum is achieved as $P \to (0,0)$.
• The sum of focal distances from any point is not fixed (unlike an ellipse) — the parabola is the degenerate case of an ellipse with one focus at infinity.
• For a focal chord with endpoints $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$:

$$P_1S + P_2S = (x_1 + a) + (x_2 + a) = x_1 + x_2 + 2a$$