Parabola Question - None

Question

The distance between a tangent to the parabola $y^2 = 4Ax$ ($A > 0$) and the parallel normal with gradient 1 is:

Accepted Answer

For parabola $y^2 = 4Ax$, a tangent with slope $m$ is $y = mx + \frac{A}{m}$.

For slope 1, tangent is: $y = x + A$

The normal with slope $-1$ (perpendicular to tangent with slope 1) has equation passing through point on parabola.

Normal at parameter $t$: $y + t(x - At^2) = 2At$

For slope $-1$: $t = 1$, giving normal $y + x - A = 2A$, or $x + y = 3A$

Distance between $y = x + A$ and $x + y = 3A$:

$$d = \frac{|3A - A|}{\sqrt{2}} = \frac{2A}{\sqrt{2}} = A\sqrt{2} = \sqrt{2A^2}$$

Rechecking: distance = $2\sqrt{2A}$

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