The ordinate of the point on the curve \( y^2=8x \) for which abscissa and ordinate change at the same rate is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
Differentiating \( y^2=8x \) with respect to time t gives the rates of change:
\( \displaystyle 2y\frac{dy}{dt}=8\frac{dx}{dt} \)
The abscissa and ordinate change at the same rate, so \( \frac{dy}{dt}=\frac{dx}{dt}\ne 0 \). Cancelling this common rate:
\( \displaystyle 2y=8 \)
\( \displaystyle y=4 \)
(The point is (2, 4), since \( x=\frac{y^2}{8}=2 \).)
Shortcut: “Same rate” means \( \frac{dy}{dx}=1 \). From \( 2y\frac{dy}{dx}=8 \), the slope is 1 where \( y=4 \).
Why the other options are incorrect
Option (1) −2 — Incorrect. At y = −2, \( 2y\frac{dy}{dt}=8\frac{dx}{dt} \) gives \( \frac{dy}{dt}=-2\frac{dx}{dt} \), not equal rates.
Option (2) −4 — Incorrect. At y = −4, \( \frac{dy}{dt}=-\frac{dx}{dt} \); the rates are equal in size but opposite in sign.
Option (4) 2 — Incorrect. At y = 2, \( \frac{dy}{dt}=2\frac{dx}{dt} \), twice the rate of the abscissa.