Question

If \( \theta \) is the acute angle between the curves \( y^2 = x \) and \( x^2 + y^2 = 2 \), then \( \tan \theta \) is:

• A. \( 1 \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( 4 \)

Hint:

The acute angle between curves is found using the slope formula. - Differentiate implicitly when dealing with equations in \( x, y \).

Answer 1

The Correct Option is B

Step 1: Compute derivatives
The slopes of the curves at the point of intersection determine \( \tan \theta \). 1st equation: \( y^2 = x \), differentiating: \[ 2y \frac{dy}{dx} = 1 \Rightarrow \frac{dy}{dx} = \frac{1}{2y}. \] 2nd equation: \( x^2 + y^2 = 2 \), differentiating: \[ 2x + 2y \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = -\frac{x}{y}. \] Step 2: Compute \( \tan \theta \)
\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|. \] Substituting slopes: \[ \tan \theta = \left| \frac{\frac{1}{2y} + \frac{x}{y}}{1 - \frac{x}{2y}} \right|. \] At \( x = 1, y = 1 \): \[ \tan \theta = \left| \frac{\frac{1}{2} + 1}{1 - \frac{1}{2}} \right| = \left| \frac{\frac{3}{2}}{\frac{1}{2}} \right| = 3. \] Thus, the correct answer is \( \boxed{3} \).