The angle between the curves `y^2=x` and `x^2=y` at (1,1) is

Options

A. `tan^(-1)(4/5)`
B. `tan^(-1)(3/4)`
C. `45^(@)`
D. `90^(@)`

Answer

To find the angle between the curves \( y^2 = x \) and \( x^2 = y \) at the point \( (1, 1) \), we will follow these steps:

### Step 1: Find the derivatives of the curves at the point (1, 1)

1. For the first curve \( y^2 = x \):
- Differentiate implicitly with respect to \( x \):
\[ 2y \frac{dy}{dx} = 1 \]
- Solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2y} \]
- Substitute \( y = 1 \):
\[ m_1 = \frac{1}{2 \cdot 1} = \frac{1}{2} \]

2. For the second curve \( x^2 = y \):
- Differentiate implicitly with respect to \( x \):
\[ 2x = \frac{dy}{dx} \]
- Substitute \( x = 1 \):
\[ m_2 = 2 \cdot 1 = 2 \]

### Step 2: Use the angle formula between two curves

The formula for the tangent of the angle \( \theta \) between two curves is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \tan \theta = \frac{\frac{1}{2} - 2}{1 + \left(\frac{1}{2}\right)(2)} \]

### Step 3: Simplify the expression

1. Calculate the numerator:
\[ \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = \frac{-3}{2} \]

2. Calculate the denominator:
\[ 1 + \left(\frac{1}{2}\right)(2) = 1 + 1 = 2 \]

3. Now substitute back:
\[ \tan \theta = \frac{\frac{-3}{2}}{2} = \frac{-3}{4} \]

### Step 4: Find the angle \( \theta \)

Since \( \tan \theta \) can be negative, we take the absolute value:
\[ \tan \theta = \frac{3}{4} \]
Thus,
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

### Conclusion

The angle between the curves \( y^2 = x \) and \( x^2 = y \) at the point \( (1, 1) \) is:
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]