Question

If a point moves along the curve $y^2=x$, At what point on the curve does the $y$ coordinate change at the same rate as the $x$ coordinate.

Answer 1

According to question,

Rate of change of $y$ = Rate of change of $x$

$⟹\frac{dy}{dt}=\frac{dx}{dt}$ ......... (1)

It is given that,

$y^2=x$

Differentiating w.r.t $t$, we get,

$2y\frac{dy}{dt}=\frac{dx}{dt}$

$⟹2y=1$ ......From eq. (1)

$⟹y=\frac{1}{2}$

By putting $y=\frac{1}{2}$, in the given eq. we get,

$⟹x=\frac{1}{4}$

Therefore the required point is $(\frac{1}{4},\frac{1}{2})$