Question

The differential equation of family of curves whose tangent form an angle of π/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)

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

Answer 2

The slope of the tangent to the family of curves is

$m_{1} = \frac{dy}{dx}$

Equation of the hyperbola is $xy = c^{2}$ ⇒ $y = \frac{c^{2}}{x}$

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

∴ Slope of tangent to $xy = c^{2}$ is $m_{2} = - \frac{c^{2}}{x^{2}}$

Now $\tan\frac{\pi}{4} = \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}}$ ⇒ $1 = \frac{\frac{dy}{dx} + \frac{c^{2}}{x^{2}}}{1 - \frac{c^{2}}{x^{2}}\frac{dy}{dx}}$

⇒ $\frac{dy}{dx}\left( 1 + \frac{c^{2}}{x^{2}} \right) = \left( 1 - \frac{c^{2}}{x^{2}} \right)$

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