Question

The two curves \( x^3 - 3x^2 + 2 = 0 \) and \( 3x y - y^3 - 2 = 0 \) intersect at an angle of:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{2} \)
• D. \( \frac{\pi}{6} \)

Hint:

To find the angle between two curves, use the formula for the tangent of the angle between the tangents at the point of intersection.

Answer 1

The Correct Option is A

Step 1: To find the angle between two curves, we first find the gradients of the tangent lines to each curve at the point of intersection.
Step 2: Use the formula for the angle between two curves: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|, \] where \( m_1 \) and \( m_2 \) are the slopes of the tangents to the curves. The angle \( \theta \) is found to be \( \frac{\pi}{4} \).

Final Answer: \[ \boxed{\frac{\pi}{4}} \]