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Q.
The differential equation of family of curves whose tangent form an angle of $\pi / 4$ with the hyperbola $x y=C^{2}$ is
Differential Equations
Solution:
Let the slope of tangent of required family be $\frac{d y}{d x}=m_{1}$
Also $y=\frac{c^{2}}{x} ;$ therefore $, \frac{d y}{d x}=-\frac{c^{2}}{x^{2}}=m_{2}($ say $)$.
By the given condition, we have $\tan \frac{\pi}{4}=\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}$
$\Rightarrow 1+m_{1} m_{2}=m_{1}-m_{2} \Rightarrow \frac{d y}{d x}+\frac{c^{2}}{x^{2}}=1-\frac{c^{2}}{x^{2}} \frac{d y}{d x}$
$\Rightarrow \frac{d y}{d x}\left(1+\frac{c^{2}}{x^{2}}\right)=1-\frac{c^{2}}{x^{2}} \Rightarrow \frac{d y}{d x}=\frac{x^{2}-c^{2}}{x^{2}+c^{2}}$