A particle moves along the curve \(y = x^{2} + 2x\). The point on the curve at which the y-coordinate is changing 6 times as fast as the x-coordinate is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (2, 8)
Both coordinates change with time t. Differentiating \(y = x^{2} + 2x\) with respect to t:
\( \displaystyle \frac{dy}{dt} = (2x + 2)\frac{dx}{dt} \)
The y-coordinate changes 6 times as fast as the x-coordinate, so \(\frac{dy}{dt} = 6\frac{dx}{dt}\):
\( 2x + 2 = 6 \)
\( x = 2 \)
Then
\( y = 4 + 4 = 8 \)
So the point is (2, 8).
Why the other options are incorrect
Option (1) (1, 3) — Incorrect. Here 2x + 2 = 4, so y changes only 4 times as fast as x.
Option (2) (3, 15) — Incorrect. Here 2x + 2 = 8, so y changes 8 times as fast as x.
Option (3) (4, 24) — Incorrect. Here 2x + 2 = 10, so y changes 10 times as fast as x.