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°

Answer 1

\[\tan^{- 1} \frac{3}{4}\]

 

\[\text { Given }: \]

\[ y^2 = x . . . \left( 1 \right)\]

\[ x^2 = y . . . \left( 2 \right)\]

\[\text { Point} = \left( 1, 1 \right)\]

\[\text { On differentiating (1) w.r.t. x, we get }\]

\[2y \frac{dy}{dx} = 1\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2y}\]

\[ \Rightarrow m_1 = \frac{1}{2}\]

\[\text { On differentiating (2) w.r.t.x, we get }\]

\[2x = \frac{dy}{dx}\]

\[ \Rightarrow m_2 = 2\left( 1 \right) = 2\]

\[\text { Now,} \]

\[\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{\frac{1}{2} - 2}{1 + \frac{1}{2} \times 2} \right| = \frac{3}{4}\]

\[ \Rightarrow \theta = \tan^{- 1} \left( \frac{3}{4} \right)\]