Question

The differential equation of family of curves whose tangent form an angle of $\pi /4$ with the hyperbola $xy=C^2$ is

• 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

Answer 1

Answer: (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 $\tan \frac{\pi }{4}=\frac{m_1-m_2}{1+m_1{m}_2}$
$\Rightarrow 1+m_1{m}_2=m_1-m_2\Rightarrow \frac{dy}{dx}+\frac{c^2}{x^2}=1-\frac{c^2}{x^2}\frac{dy}{dx}$
$\Rightarrow \frac{dy}{dx}\left(1+\frac{c^2}{x^2}\right)=1-\frac{c^2}{x^2}\Rightarrow \frac{dy}{dx}=\frac{x^2-c^2}{x^2+c^2}$