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Q.
Acute angle between two curves $x^{2}+y^{2}=a^{2} \sqrt{2}$ and $x^{2}-y^{2}=a^{2}$ is
Application of Derivatives
Solution:
$x^{2}+y^{2}=a^{2} \sqrt{2}$ 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{2 x y}{y^{2}-x^{2}}|$
Squaring and subtracting the equations of given curves,
$ 4 x^{2} y^{2}=a^{4}$
$\therefore 2 x y=\pm a^{2} $
$ \therefore \theta =\left|\tan ^{-1} \frac{a^{2}}{a^{2}}\right| $
$=\tan ^{-1}(1)$
$=\frac{\pi}{4}$