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Question: The angle between the curves \(y^{2} = x\) and \(x^{2} = y\) at (1, 1) is...

The angle between the curves y2=xy^{2} = x and x2=yx^{2} = y at (1, 1) is

A

tan⁡−143\tan^{- 1}\frac{4}{3}

B

tan⁡−134\tan^{- 1}\frac{3}{4}

C

90o90^{o}

D

45o45^{o}

Answer

tan⁡−134\tan^{- 1}\frac{3}{4}

Explanation

Solution

Given curve y2=xy^{2} = x and x2=yx^{2} = y

Differentiating w.r.t. x, 2ydydx=12y\frac{dy}{dx} = 1 and 2x=dydx2x = \frac{dy}{dx}

(dydx)(1,1)=12\left( \frac{dy}{dx} \right)_{(1,1)} = \frac{1}{2} and (dydx)(1,1)=2\left( \frac{dy}{dx} \right)_{(1,1)} = 2

Angle between the curve

⇒ tan⁡φ=2−121+12.2\tan\varphi = \frac{2 - \frac{1}{2}}{1 + \frac{1}{2}.2} ⇒ tan⁡φ=34\tan\varphi = \frac{3}{4} ⇒φ=tan⁡−134\varphi = \tan^{- 1}\frac{3}{4}.