Question

A particle moves on the curve y= x³.Find the points on the curve at which the y-coordinate changes w.r.t. time thrice as fast as x-coordinate.

Answer 1

A particle moves on the curve, y = x³

differentiating both sides with respect to time,

i.e., (dy/dt) = 3x² (dx/dt) ......(1)

a/c to question,

y- coordinate changes w.r.t thrice as fast as x - coordinate.

i.e., (dy/dt) = 3 (dx/dt)......(2)

from equations (1) and (2),

⇒3(dx/dt) = 3x²(dx/dt)

⇒3 = 3x²

⇒1 = x²

⇒x = ±1

so, y = x³ = (±1)³ = ± 1

hence, there are two points on the curve at which the y-coordinate changes w.r.t. time thrice as fast as x-coordinate.

.these are ; (1, 1) and (-1, -1)