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Question
The angle between the curves y2 = x and x2 = y at (1, 1) is ______________ .
Options
\[\tan^{- 1} \frac{4}{3}\]
\[\tan^{- 1} \frac{3}{4}\]
90°
45°
Solution
\[\tan^{- 1} \frac{3}{4}\]
\[\text { Given }: \]
\[ y^2 = x . . . \left( 1 \right)\]
\[ x^2 = y . . . \left( 2 \right)\]
\[\text { Point} = \left( 1, 1 \right)\]
\[\text { On differentiating (1) w.r.t. x, we get }\]
\[2y \frac{dy}{dx} = 1\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2y}\]
\[ \Rightarrow m_1 = \frac{1}{2}\]
\[\text { On differentiating (2) w.r.t.x, we get }\]
\[2x = \frac{dy}{dx}\]
\[ \Rightarrow m_2 = 2\left( 1 \right) = 2\]
\[\text { Now,} \]
\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{\frac{1}{2} - 2}{1 + \frac{1}{2} \times 2} \right| = \frac{3}{4}\]
\[ \Rightarrow \theta = \tan^{- 1} \left( \frac{3}{4} \right)\]
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