Question

The differential equation of family of curves whose tangent form an angle of $\frac{\pi }{4}$ with the hyperbola $xy=c^2$

• 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)

$\frac{dy}{dx}=\frac{x^2-c^2}{x^2+c^2}$

Let the slope of the tangent of family of curve be $m_1=\frac{dy}{dx}$
Given hyperbola $xy=c^2$
$\Rightarrow y=\frac{c^2}{x}$
$\Rightarrow \frac{dy}{dx}=-\frac{c^2}{x^2}=m_2\left(say\right)$
According to given condition ,
$\tan \frac{\pi }{4}=\frac{m_1-m_2}{1+m_1{m}_2}$
$\Rightarrow 1=\frac{m_1-m_2}{1+m_1{m}_2}$
$\Rightarrow 1+m_1{m}_2=m_1-m_2$
$\Rightarrow 1-\frac{dy}{dx}\frac{c^2}{x^2}=\frac{dy}{dx}+\frac{c^2}{x^2}$
$\Rightarrow (1+\frac{c^2}{x^2})\frac{dy}{dx}=1-\frac{c^2}{x^2}$
$\Rightarrow \frac{dy}{dx}=\frac{x^2-c^2}{x^2+c^2}$