Condition of Tangency for a Parabola

The question: for which values of $m$ and $c$ does the general line $y=mx+c$ touch (rather than cross, or miss entirely) the parabola $y^2=4ax$? And if it does touch, at what point?

Write the line as $mx-y+c=0$ … (I).

We already know (section 7.1.11) that the tangent at a specific point $P(x_1,y_1)$ is $yy_1=2a(x+x_1)$, which rearranges to

If the line (I) actually IS the tangent at some point $(x_1,y_1)$, then (I) and (II) must be the same line — so their coefficients must be proportional:

From the first equality with the second: $y_1=\dfrac{2a}{m}$. From the first with the third: $x_1=\dfrac{c}{m}$.

But $(x_1,y_1)$ must actually lie on the parabola, so $y_1^2=4ax_1$:

So the condition of tangency is $\boxed{c=\dfrac{a}{m}}$ — the line $y=mx+c$ touches $y^2=4ax$ if and only if $c=a/m$, and the point of contact is

Equivalently, the tangent to $y^2=4ax$ with a given slope $m$ can always be written directly as $y=mx+\dfrac{a}{m}$.

Worked Example 1 — tangent at a given point. Parabola $y^2=9x$ ($4a=9\Rightarrow a=\dfrac94$); tangent at $(1,-3)$: using $yy_1=2a(x+x_1)$: $y(-3)=2\left(\dfrac94\right)(x+1) \Rightarrow -3y=\dfrac92(x+1) \Rightarrow -6y=9(x+1) \Rightarrow 9x+6y+9=0 \Rightarrow 3x+2y+3=0$. …