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 $(2,4)$.