Question

A particle moves along the curve y = x³. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-coordinate.

Answer 1

\[\text { According to the question },\]

\[\frac{dy}{dt} = 3\frac{dx}{dt}\]

\[\text { Now,} \]

\[y = x^3 \]

\[ \Rightarrow \frac{dy}{dt} = 3 x^2 \frac{dx}{dt}\]

\[ \Rightarrow 3\frac{dx}{dt} = 3 x^2 \frac{dx}{dt}\]

\[ \Rightarrow x^2 = 1\]

\[ \Rightarrow x = \pm 1\]

\[\text { Substituting x }=\pm1 \text { in y }= x^3 , \text { we get }\]

\[y = \pm 1\]

\[\text { So the points are }\left( 1, 1 \right)\text { and }\left( - 1, - 1 \right).\]