If \(\sqrt{x} + \sqrt{y} = 5\), then the value of \(\displaystyle\frac{dy}{dx}\) at the point (4, 9) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle -\frac{3}{2}\)
The point lies on the curve, since \(\sqrt{4} + \sqrt{9} = 2 + 3 = 5\). By implicit differentiation with respect to x:
\( \displaystyle \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}}\,\frac{dy}{dx} = 0 \)
\( \displaystyle \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \)
At (4, 9):
\( \displaystyle \frac{dy}{dx} = -\frac{3}{2} \)
Why the other options are incorrect
Option (1) \(\displaystyle -\frac{2}{3}\) — Incorrect. This is \(-\frac{\sqrt{x}}{\sqrt{y}}\), the ratio turned upside down.
Option (2) \(\displaystyle \frac{3}{2}\) — Incorrect. The minus sign is lost; as x increases, \(\sqrt{y}\) must decrease to keep the sum 5.
Option (3) \(\displaystyle -\frac{9}{4}\) — Incorrect. This is \(-\frac{y}{x}\); the square roots have been dropped.