Applications Of Derivatives Question 424
Question: The number of tangents to the curve $ {x^{3/2}}+{y^{3/2}}=2{a^{3/2}},,a>0, $ which are equally inclined to the axes, is
Options:
A) 2
B) 1
C) 0
D) 4
Show Answer
Answer:
Correct Answer: B
Solution:
[b] Given curve is $ {x^{3/2}}+{y^{3/2}}+{y^{3/2}}=2{a^{3/2}}…(1) $
$ \therefore \frac{3}{2}\sqrt{x}+\frac{3}{2}\sqrt{y}\frac{dy}{dx}=0 $ or $ \frac{dy}{dx}=-\frac{\sqrt{x}}{\sqrt{y}} $
Since the tangent is equally inclined to axes, $ \frac{dy}{dx}=\pm 1\therefore -\frac{\sqrt{x}}{\sqrt{y}}=\pm 1or-\frac{\sqrt{x}}{\sqrt{y}}=-1 $
$ \therefore \sqrt{x}=\sqrt{y}[\because \sqrt{x}>0,\sqrt{y}>0] $ Putting $ \sqrt{y}=\sqrt{x} $ in (1), we get
$ 2{x^{3/2}}=2{a^{3/2}},or,x^{3}=a^{3} $ . Therefore, $ x=a $ and, so, $ y=a $ .
BETA
