Show that the line $2x - y + 2 = 0$ is a tangent to the parabola $y^2 = 16x$.

Question

Accepted Answer

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> [!TLDR]
> Write the line as $y=mx+c$ and check the standard tangency condition $c=a/m$ for the parabola $y^2=4ax$.
>
> The parabola $y^2=16x$ is of the form $y^2=4ax$ with $4a=16 \Rightarrow a=4$.
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> A line $y=mx+c$ touches $y^2=4ax$ if and only if $c=\dfrac{a}{m}$.
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> Rewrite $2x-y+2=0$ as $y=2x+2$, so $m=2$, $c=2$.
>
> Check: $\dfrac{a}{m} = \dfrac{4}{2} = 2 = c$.
> …

Equation	Opens	Focus	Directrix
$y^2 = 4ax$	right	$(a, 0)$	$x = -a$
$y^2 = -4ax$	left	$(-a, 0)$	$x = a$
$x^2 = 4ay$	up	$(0, a)$	$y = -a$
$x^2 = -4ay$	down	$(0, -a)$	$y = a$

Q.Show that the line $2x - y + 2 = 0$ is a tangent to the parabola $y^2 = 16x$ .

Concept understanding — Parabola Standard Form

Standard Equations of a Parabola

The four standard forms

Where $y^2 = 4ax$ comes from

Latus rectum

Worked example

Q.Show that the line 2x−y+2=02x - y + 2 = 02x−y+2=0 is a tangent to the parabola y2=16xy^2 = 16xy2=16x.

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Q.Show that the line $2x - y + 2 = 0$ is a tangent to the parabola $y^2 = 16x$ .

Unlock everything free for 14 days