Find the point on the curve

Question:

Find the point on the curve $y^{2}=8 x$ for which the abscissa and ordinate change at the same rate.

Solution:

Here,

$y^{2}=8 x \quad \ldots(1)$

$\Rightarrow 2 y \frac{d y}{d t}=8 \frac{d x}{d t}$

$\Rightarrow 2 y=8$         $\left[\because \frac{d y}{d t}=\frac{d x}{d t}\right]$

$\Rightarrow y=4$

$\Rightarrow x=\frac{y^{2}}{8}$      [From eq. (1)]

$\Rightarrow x=\frac{16}{8}=2$

So, the point is $(2,4)$.

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