The two curves `x^(3)-3xy^(2)+5=0` and `3x^(2)y-y^(3)-7=0`

Options

A. cut at right angle
B. touch each other
C. cut at an angle`(pi)/(4)`
D. cut at an angle `(pi)/(3)`

Explanation

To find the angle between the two curves given by the equations \( x^3 - 3xy^2 + 5 = 0 \) and \( 3x^2y - y^3 - 7 = 0 \), we will follow these steps:

### Step 1: Differentiate the first curve

The first curve is given by:
\[ C_1: x^3 - 3xy^2 + 5 = 0 \]

We differentiate this implicitly with respect to \( x \):
\[ \frac{d}{dx}(x^3) - \frac{d}{dx}(3xy^2) + \frac{d}{dx}(5) = 0 \]

Using the product rule on \( 3xy^2 \):
\[ 3(1 \cdot y^2 + x \cdot 2y \frac{dy}{dx}) = 3y^2 + 6xy\frac{dy}{dx} \]

So, the differentiation gives us:
\[ 3x^2 - (3y^2 + 6xy\frac{dy}{dx}) = 0 \]

Rearranging gives:
\[ 3x^2 - 3y^2 = 6xy\frac{dy}{dx} \]

Thus, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{x^2 - y^2}{2xy} \]

### Step 2: Differentiate the second curve

The second curve is given by:
\[ C_2: 3x^2y - y^3 - 7 = 0 \]

Differentiating this implicitly:
\[ \frac{d}{dx}(3x^2y) - \frac{d}{dx}(y^3) - \frac{d}{dx}(7) = 0 \]

Using the product rule on \( 3x^2y \):
\[ 3(2xy + x^2\frac{dy}{dx}) - 3y^2\frac{dy}{dx} = 0 \]

This leads to:
\[ 6xy + 3x^2\frac{dy}{dx} - 3y^2\frac{dy}{dx} = 0 \]

Rearranging gives:
\[ 6xy = (3y^2 - 3x^2)\frac{dy}{dx} \]

Thus, we can express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{6xy}{3y^2 - 3x^2} = \frac{2xy}{y^2 - x^2} \]

### Step 3: Find the slopes of the curves

Let:
- \( m_1 = \frac{x^2 - y^2}{2xy} \) (slope of the first curve)
- \( m_2 = \frac{2xy}{y^2 - x^2} \) (slope of the second curve)

### Step 4: Determine the angle between the curves

The curves are perpendicular if:
\[ m_1 \cdot m_2 = -1 \]

Calculating \( m_1 \cdot m_2 \):
\[ m_1 \cdot m_2 = \left(\frac{x^2 - y^2}{2xy}\right) \cdot \left(\frac{2xy}{y^2 - x^2}\right) \]
\[ = \frac{(x^2 - y^2)(2xy)}{2xy(y^2 - x^2)} \]
\[ = \frac{(x^2 - y^2)}{(y^2 - x^2)} = -1 \]

Since \( m_1 \cdot m_2 = -1 \), the curves intersect at a right angle.

### Conclusion

The two curves intersect at right angles.