If \( x=\sqrt{a^{\tan^{-1}t}} \) and \( y=\sqrt{a^{\cot^{-1}t}} \) then which one of the following is true?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( xdy+ydx=0 \)
The product xy does not depend on t, by the identity \( \tan^{-1}t+\cot^{-1}t=\frac{\pi}{2} \):
\( \displaystyle xy=\sqrt{a^{\tan^{-1}t}\cdot a^{\cot^{-1}t}} \)
\( \displaystyle =\sqrt{a^{\tan^{-1}t+\cot^{-1}t}} \)
\( \displaystyle =\sqrt{a^{\pi/2}} \)
So xy is a constant. Differentiating xy = constant by the product rule:
\( \displaystyle x\,dy+y\,dx=0 \)