Question

Acute angle between two curves $x^2+y^2=a^2\sqrt{2}$ and $x^2-y^2=a^2$ is

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. none of these

Answer 1

Answer: (C)

$x^2+y^2=a^2\sqrt{2}$ and $x^2-y^2=a^2$
$\therefore \frac{dy}{dx}=\frac{-x}{y};\frac{dy}{dx}=\frac{x}{y}$
Angle between curves is given by
$\theta =\left|{\tan }^{-1}\frac{\frac{x}{y}+\frac{x}{y}}{1-\frac{x^2}{y^2}}\right|$
$|{\tan }^{-1}\frac{2xy}{y^2-x^2}|$
Squaring and subtracting the equations of given curves,
$4x^2{y}^2=a^4$
$\therefore 2xy=\pm a^2$
$\therefore \theta =\left|{\tan }^{-1}\frac{a^2}{a^2}\right|$
$={\tan }^{-1}(1)$
$=\frac{\pi }{4}$