Applications Of Derivatives Question 429
Question: The straight line $ \frac{x}{a}+\frac{y}{b}=2 $ touches the curve $ {{( \frac{x}{a} )}^{n}}+{{( \frac{y}{b} )}^{n}}=2 $ at the point (a, b) for
Options:
A) n = 1, 2
B) n = 3, 4, -5
C) n = 1, 2, 3
D) Any value of n
Show Answer
Answer:
Correct Answer: D
Solution:
[d] The point (a, b) lies on both the straight line and the given curve $ {{( \frac{x}{a} )}^{n}}+{{( \frac{y}{b} )}^{n}}=2 $ .
Differentiating the equation, we get $ \frac{dy}{dx}=-\frac{{x^{n-1}}}{a^{n}}.\frac{b^{n}}{{y^{n-1}}} $
$ \therefore {{( \frac{dy}{dx} )}_{at(a,b)}}=-\frac{b}{a}= $ the slope of $ \frac{x}{a}+\frac{y}{b}=2 $
Hence, it touches the curve at (a, b) whatever may be the value of n.
BETA
