The angle between `y^(2)=x and x^(2)=y` at the origin is

Options

A. `2 tan^(-1)""(3)/(4)`
B. `tan^(-1)""(4)/(3)`
C. `(pi)/(2)`
D. `(pi)/(4)`

Answer

To find the angle between the curves given by the equations \(y^2 = x\) and \(x^2 = y\) at the origin, we will follow these steps:

### Step 1: Differentiate the equations
We need to find the slopes of the tangents to the curves at the origin.

1. For the curve \(y^2 = x\):
- Differentiate both sides with respect to \(x\):
\[ 2y \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{2y} \]

2. For the curve \(x^2 = y\):
- Differentiate both sides with respect to \(x\):
\[ 2x = \frac{dy}{dx} \implies \frac{dy}{dx} = 2x \]

### Step 2: Evaluate the slopes at the origin
Now we will evaluate the slopes at the origin \((0, 0)\).

1. For \(y^2 = x\):
- At the origin, \(y = 0\):
\[ \frac{dy}{dx} = \frac{1}{2(0)} \quad \text{(undefined)} \]

2. For \(x^2 = y\):
- At the origin, \(x = 0\):
\[ \frac{dy}{dx} = 2(0) = 0 \]

### Step 3: Determine the angle between the curves
Let \(m_1\) and \(m_2\) be the slopes of the two curves at the origin. Since \(m_1\) is undefined (which corresponds to a vertical line) and \(m_2 = 0\) (which corresponds to a horizontal line), we can use the formula for the angle \(\theta\) between two lines given their slopes:

\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

In this case, since \(m_1\) is undefined, we can conclude that the angle between the vertical and horizontal lines is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

### Conclusion
Thus, the angle between the curves \(y^2 = x\) and \(x^2 = y\) at the origin is:

\[ \theta = \frac{\pi}{2} \text{ radians} \quad \text{(or 90 degrees)} \]