Applications Of Derivatives Question 46
Question: If a variable tangent to the curve $ x^{2}y=c^{3} $ makes intercepts a and b on x-and y-axis, respectively, then the value of $ a^{2}b $ is
Options:
A) 27 $ c^{3} $
B) $ \frac{4}{27}c^{3} $
C) $ \frac{27}{4}c^{3} $
D) $ \frac{4}{9}c^{3} $
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ x^{2}y=c^{3} $ Differentiating w.r.t. x, we have $ x^{2}\frac{dy}{dx}+2xy=0 $ or $ \frac{dy}{dx}=-\frac{2y}{x} $ Equation of the tangent at (h, k) is $ y-k=-\frac{2k}{h}(x-h) $
$ y=0 $ gives $ x=\frac{3h}{2}=a $ , and x=0 gives $ y=3k=b $ . Now, $ a^{2}b=\frac{9h^{2}}{4}3k=\frac{27}{4}h^{2}k=\frac{27}{4}{c^{3.}} $
BETA
