Find equation of tangent to parabola y²=8x at point (2,4)

Question

Find the equation of the tangent to the parabola $y^2 = 8x$ at the point $(2, 4)$ .

Solution — Step by Step

The standard form of a parabola is $y^2 = 4ax$ . Comparing with $y^2 = 8x$ :

4a = 8 \implies a = 2

So this parabola has focus at $(2, 0)$ and directrix $x = -2$ .

We can verify that $(2, 4)$ lies on the parabola: $4^2 = 16$ and $8 \times 2 = 16$ ✓

For the parabola $y^2 = 4ax$ , the equation of the tangent at point $(x_1, y_1)$ is:

yy_1 = 2a(x + x_1)

Here $a = 2$ , $x_1 = 2$ , $y_1 = 4$ . Substituting:

y(4) = 2(2)(x + 2)

4y = 4(x + 2)

y = x + 2

The equation of the tangent at $(2, 4)$ is:

\boxed{y = x + 2}

or equivalently, $x - y + 2 = 0$ .

Why This Works

The tangent formula $yy_1 = 2a(x + x_1)$ is derived using implicit differentiation. Differentiating $y^2 = 4ax$ with respect to $x$ :

2y\frac{dy}{dx} = 4a \implies \frac{dy}{dx} = \frac{2a}{y}

At $(2, 4)$ : slope $= \frac{2 \times 2}{4} = 1$ .

Using point-slope form: $y - 4 = 1(x - 2) \implies y = x + 2$ .

Both methods — the formula and implicit differentiation — give the same result. The formula is faster for board exams; the differentiation approach shows you understand the concept.

Alternative Method

Using implicit differentiation from scratch:

$y^2 = 8x$

Differentiate both sides with respect to $x$ :

2y\frac{dy}{dx} = 8 \implies \frac{dy}{dx} = \frac{4}{y}

At point $(2, 4)$ : slope $m = 4/4 = 1$ .

Tangent: $y - 4 = 1(x - 2) \implies y = x + 2$ .

The tangent formula $T = 0$ for a parabola is formed by replacing $y^2$ with $yy_1$ and $x$ with $\frac{x + x_1}{2}$ in the parabola equation. This is a systematic “T = 0” rule valid for all conics — learn it once and apply to circle, parabola, ellipse, hyperbola all in the same way.

Common Mistake

Students sometimes use the normal formula instead of the tangent formula, or forget the " $x + x_1$ " form. The tangent at point $(x_1, y_1)$ on $y^2 = 4ax$ is $yy_1 = 2a(x + x_1)$ — note it’s $x + x_1$ (not $x - x_1$ ). This asymmetry catches many students off guard. Always derive it from the slope if you forget.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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