Question

A particle moves along the curve $6y = x^3 + 2$. The point $?P?$ on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate, are $(4, 11)$ and $\left(-4, -\frac{31}{3}\right).$

• A. x-coordinates at the point P are $? 4$
• B. y-coordinates at the point P are 11 and $\frac{-31}{3}$
• C. Both (a) and (b)
• D. None of the above

Answer 1

Answer: (C)

Both (a) and (b)

Answer 2

Given, $6y = x^3 + 2$ On differentiating w.r.t. t, we get $6 \frac{dy}{dt}=3x^{2} \frac{dx}{dt} \Rightarrow 6\times8 \frac{dx}{dt}=3x^{2} \frac{dx}{dt}$ $\Rightarrow 3x^{2}=48 \Rightarrow x^{2}=16$ $\Rightarrow x=\pm4$ When $x = 4$, then $6y = \left(4\right)^{3} + 2$ $\Rightarrow 6y=64+2 \Rightarrow y=\frac{66}{6}=11$ When $x = - 4$, then $6y = \left(- 4\right)^{3} + 2$ $\Rightarrow 6y=-64+2$ $\Rightarrow y=\frac{-62}{6}=\frac{-31}{3}$ Hence, the required points on the curve are $\left(4, 11\right)$ and $\left(-4, \frac{-31}{3}\right)$