Question

The angle between \( x^2 = y \) and \( y^2 = x \) at \( (1, 1) \) is:

• A. $\tan^{-1} \left( \frac{3}{4} \right)$
• B. $\tan^{-1} \left( \frac{4}{3} \right)$
• C. $\tan^{-1} \left( \frac{1}{3} \right)$
• D. $\tan^{-1} \left( \frac{1}{4} \right)$

Hint:

The angle between two curves at their point of intersection is the angle between their tangents at that point.

Answer 1

The Correct Option is A

Step 1: Find the slopes of the tangents to the curves at the point \( (1, 1) \).
For \( x^2 = y \), $\frac{dy}{dx} = 2x$. At \( (1, 1) \), $m_1 = 2(1) = 2$.
For \( y^2 = x \), $2y \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{2y}$. At \( (1, 1) \), $m_2 = \frac{1}{2(1)} = \frac{1}{2}$. Step 2: Use the formula for the angle between two lines.
$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{2 - \frac{1}{2}}{1 + 2 \times \frac{1}{2}} \right| = \left| \frac{\frac{3}{2}}{2} \right| = \frac{3}{4}$. Step 3: Find the angle \( \theta \).
$\theta = \tan^{-1} \left( \frac{3}{4} \right)$.