Differentiation Question 403
Question: If $ y^{2}=ax^{2}+bx+c $ , then $ y^{3}\frac{d^{2}y}{dx^{2}} $ is
[DCE 1999]
Options:
A) A constant
B) A function of x only
C) A function of y only
D) A function of x and y
Show Answer
Answer:
Correct Answer: A
Solution:
$ y^{2}=ax^{2}+bx+c\Rightarrow 2y\frac{dy}{dx}=2ax+b $
Therefore $ 2{{( \frac{dy}{dx} )}^{2}}+2y\frac{d^{2}y}{dx^{2}}=2a\Rightarrow y\frac{d^{2}y}{dx^{2}}=a-{{( \frac{dy}{dx} )}^{2}} $
Therefore $ y\frac{d^{2}y}{dx^{2}}=a-{{( \frac{2ax+b}{2y} )}^{2}} $
Therefore $ y\frac{d^{2}y}{dx^{2}}=\frac{4ay^{2}-{{(2ax+b)}^{2}}}{4y^{2}} $
Therefore $ 4y^{3}\frac{d^{2}y}{dx^{2}}=4a(ax^{2}+bx+c)-(4a^{2}x^{2}+4abx+b^{2}) $
Therefore $ 4y^{3}\frac{d^{2}y}{dx^{2}}=4ac-b^{2}\Rightarrow y^{3}\frac{d^{2}y}{dx^{2}}=\frac{4ac-b^{2}}{4}= $ a constant.
BETA
