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The Angle between the parabolas $y^{2}=x$ and $x^{2}=y$ at the origin is

\[\begin{array}{1 1}(1)2\tan^{-1}\left( \begin{array}{c} 3 \\ 4 \end{array} \right)&(2)\tan^{-1}\left( \begin{array}{c} 4 \\ 3 \end{array} \right)\\(3)\frac{\pi}{2}&(4)\frac{\pi}{4}\end{array}\]

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$y^2=x$
$y=x^2$
$2y. \large\frac{dy}{dx} $$=1=> \large\frac{dy}{dx}$$=2x$
$ \large\frac{dy}{dx} =\frac{1}{2y}=> \large\frac{dy}{dx}$$=2x$
$\large\frac{dy}{dx} $$ /(0,0) =\frac{1}{0} $$= \infty $
$\large\frac{dy}{dx}$$ /(0,0)=0$
At the point (0,0) one tangent is parallel to x-axis ad the other tangent is perpendicular to x-axis .
Angle between the curves at (0,0) is the angle between the tangent at (0,0)
$=> \large\frac{\pi}{2}$
Hence 3 is the correct answer.
answered May 19, 2014 by meena.p
 

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