The curve `x^3 - 3xy^2 + 2 = 0` and `3x^2 y-y^3 -2= 0` cut at an angle of

Options

A. `45^@`
B. `60^@`
C. `90^@`
D. `30^@`

Answer

To find the angle at which the curves \( x^3 - 3xy^2 + 2 = 0 \) and \( 3x^2y - y^3 - 2 = 0 \) intersect, we need to follow these steps:

### Step 1: Find the Points of Intersection
To find the points of intersection, we need to solve the two equations simultaneously.

1. From the first equation:
\[ x^3 - 3xy^2 + 2 = 0 \quad \text{(1)} \]
2. From the second equation:
\[ 3x^2y - y^3 - 2 = 0 \quad \text{(2)} \]

We can solve these equations for \(x\) and \(y\).

### Step 2: Differentiate Each Curve
Next, we need to find the slopes of the tangents to the curves at the points of intersection.

1. For the first curve, differentiate equation (1):
\[ \frac{dy}{dx} = \frac{-\frac{d}{dx}(x^3 - 2)}{\frac{d}{dx}(-3xy^2)} = \frac{-3x^2}{-3y^2 - 6xy\frac{dy}{dx}} \]
Rearranging gives:
\[ \frac{dy}{dx} = \frac{3x^2}{3y^2 + 6xy\frac{dy}{dx}} \]

2. For the second curve, differentiate equation (2):
\[ \frac{dy}{dx} = \frac{-\frac{d}{dx}(3x^2y - 2)}{\frac{d}{dx}(-y^3)} = \frac{-3(2xy + x^2\frac{dy}{dx})}{-3y^2\frac{dy}{dx}} \]
Rearranging gives:
\[ \frac{dy}{dx} = \frac{3(2xy)}{3y^2 - 3x^2\frac{dy}{dx}} \]

### Step 3: Evaluate Slopes at Intersection Points
Let the slopes at the intersection point \( (a, b) \) be \( M_1 \) and \( M_2 \).

1. For the first curve, substituting \( (a, b) \):
\[ M_1 = \frac{3a^2}{3b^2 + 6abM_1} \]

2. For the second curve, substituting \( (a, b) \):
\[ M_2 = \frac{3(2ab)}{3b^2 - 3a^2M_2} \]

### Step 4: Use the Angle Between Two Lines Formula
The angle \( \theta \) between the two tangents can be found using the formula:
\[ \tan(\theta) = \left| \frac{M_1 - M_2}{1 + M_1M_2} \right| \]

### Step 5: Determine the Condition for Perpendicularity
If the curves intersect at right angles, then:
\[ M_1 \cdot M_2 = -1 \]
This condition indicates that the angle between the two tangents is \( 90^\circ \) or \( \frac{\pi}{2} \).

### Conclusion
After evaluating the slopes and applying the condition for perpendicularity, we find that the curves intersect at an angle of \( \frac{\pi}{2} \) radians (90 degrees).