Question

The two curves \( x^3 - 3xy^2 + 2 = 0 \) and \( 3x^2y - y^3 = 2 \) are

• A. cut at right angles
• B. touch each other
• C. cut at an angle \( \frac{\pi}{3} \)
• D. cut at an angle \( \frac{\pi}{4} \)

Hint:

To determine if curves cut at right angles, calculate the gradients of the curves at their point of intersection and use the formula for the angle between two curves.

Answer 1

The Correct Option is A

Step 1: Understand the given curves.
We are given two curves: 1. \( x^3 - 3xy^2 + 2 = 0 \)
2. \( 3x^2y - y^3 = 2 \)
We need to determine whether these curves cut at right angles, touch each other, or cut at some other angle.

Step 2: Calculate the gradients of the curves.
The angle between two curves at their point of intersection can be determined by calculating the gradients (slopes) of the tangents to the curves at the intersection point. - For the first curve, \( f(x, y) = x^3 - 3xy^2 + 2 \), the gradient is given by the partial derivatives: \[ \frac{\partial f}{\partial x} = 3x^2 - 3y^2 \quad \text{and} \quad \frac{\partial f}{\partial y} = -6xy \] - For the second curve, \( g(x, y) = 3x^2y - y^3 = 2 \), the gradient is: \[ \frac{\partial g}{\partial x} = 6xy \quad \text{and} \quad \frac{\partial g}{\partial y} = 3x^2 - 3y^2 \]

Step 3: Calculate the angle between the curves.
The angle \( \theta \) between the two curves is given by the formula: \[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] where \( m_1 \) and \( m_2 \) are the gradients of the tangents to the curves at the point of intersection. After calculating the gradients and substituting into the formula, we find that \( \theta = 90^\circ \), meaning the curves cut at right angles.

Step 4: Conclusion.
Thus, the two curves cut at right angles, corresponding to option (A).

Step 5: Verification.
By verifying the gradients and calculating the angle, we confirm that the curves indeed intersect at right angles.