Differential Equations Question 57
Question: The differential equation of family of curves whose tangent form an angle of $ \pi /4 $ with the hyperbola $ xy=C^{2} $ is
Options:
A) $ \frac{dy}{dx}=\frac{x^{2}+C^{2}}{x^{2}-C^{2}} $
B) $ \frac{dy}{dx}=\frac{x^{2}-C^{2}}{x^{2}+C^{2}} $
C) $ \frac{dy}{dx}=-\frac{C^{2}}{x^{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
[b] Let the slope of tangent of required family be $ \frac{dy}{dx}=m_1 $ Also $ y=\frac{C^{2}}{x} $ ; therefore, $ \frac{dy}{dx}=-\frac{C^{2}}{x^{2}}=m^{2} $ (say). By the given condition, we have $ \tan \frac{\pi }{4} $
$ =\frac{m_1-m_2}{1+m_1m_2}\Rightarrow 1+m_1m_2=m_1-m_2 $
$ \Rightarrow \frac{dy}{dx}+\frac{C^{2}}{x^{2}}=1-\frac{C^{2}}{x^{2}}\frac{dy}{dx}\Rightarrow \frac{dy}{dx}( 1+\frac{C^{2}}{x^{2}} ) $
$ =1-\frac{C^{2}}{x^{2}}\Rightarrow \frac{dy}{dx}=\frac{x^{2}-C^{2}}{x^{2}+C^{2}} $
BETA
