Condition for a Line to be Tangent

Line y = mx + c is tangent to y² = 4ax if and only if c = a/m. Unlike circles, only one value of c works for each slope m.

The condition $c = a/m$ is derived fully in the tangent-slope derivation. Here the focus is on its geometric meaning and application.

Geometric interpretation: $c = a/m$ means that among all lines with slope $m$, exactly one is tangent to $y^2 = 4ax$. The perpendicular distance from the focus $S(a, 0)$ to this line equals the distance from $S$ to the directrix — but the parabola condition is more cleanly stated as $c = a/m$ than as a distance equation.

Test: Is $y = 2x + 3$ tangent to $y^2 = 8x$? Here $m = 2$, $a = 2$, $c = 3$. Condition: $c = a/m = 2/2 = 1 \neq 3$. Not tangent.

Is $y = 2x + 1$ tangent? $c = 1 = a/m = 1$. Yes, tangent. Contact point: $(a/m^2, 2a/m) = (1/2, 2)$.

For $x^2 = 4ay$: The line $y = mx + c$ is tangent iff $c = -am^2$. (Derived by substituting $y = mx + c$ into $x^2 = 4ay$ and setting discriminant to zero.)