Three normal are drawn to the curve `y^(2)=x` from a point (c, 0) . Out of three, one is always the x-axis. If two other normal are perpendicular to each other, then ‘c’ is:

Options

A. `(3)/(4)`
B. `(1)/(2)`
C. `(3)/(2)`
D. 2

Explanation

To solve the problem, we need to find the value of \( c \) such that three normals drawn from the point \( (c, 0) \) to the curve \( y^2 = x \) include the x-axis and two other normals that are perpendicular to each other.

### Step-by-Step Solution

1. **Understand the curve**: The given curve is \( y^2 = x \), which is a parabola that opens to the right.

2. **Equation of the normal**: The equation of the normal to the parabola \( y^2 = 4ax \) at a point \( (at^2, 2at) \) is given by:
\[ y = mx - 2am - am^3 \]
where \( m \) is the slope of the normal.

3. **Identify \( a \)**: For the curve \( y^2 = x \), we can rewrite it in the standard form \( y^2 = 4ax \). Here, comparing, we find:
\[ 4a = 1 \implies a = \frac{1}{4} \]

4. **Substituting \( a \)**: Substitute \( a = \frac{1}{4} \) into the normal equation:
\[ y = mx - 2\left(\frac{1}{4}\right)m - \left(\frac{1}{4}\right)m^3 \]
Simplifying this gives:
\[ y = mx - \frac{1}{2}m - \frac{1}{4}m^3 \]

5. **Normal from point \( (c, 0) \)**: Since the normal passes through the point \( (c, 0) \), we set \( y = 0 \):
\[ 0 = mc - \frac{1}{2}m - \frac{1}{4}m^3 \]
Rearranging gives:
\[ \frac{1}{4}m^3 - \left(c - \frac{1}{2}\right)m = 0 \]

6. **Factoring the equation**: Factoring out \( m \):
\[ m\left(\frac{1}{4}m^2 - \left(c - \frac{1}{2}\right)\right) = 0 \]
This gives us one root \( m = 0 \) (the normal along the x-axis) and the other roots from:
\[ \frac{1}{4}m^2 - \left(c - \frac{1}{2}\right) = 0 \]

7. **Finding the other slopes**: Solving for \( m^2 \):
\[ m^2 = 4\left(c - \frac{1}{2}\right) \]
Thus, the slopes are:
\[ m_2 = \sqrt{4\left(c - \frac{1}{2}\right)}, \quad m_3 = -\sqrt{4\left(c - \frac{1}{2}\right)} \]

8. **Condition for perpendicularity**: The two normals are perpendicular, which means:
\[ m_2 \cdot m_3 = -1 \]
Substituting the values:
\[ \sqrt{4\left(c - \frac{1}{2}\right)} \cdot -\sqrt{4\left(c - \frac{1}{2}\right)} = -1 \]
This simplifies to:
\[ -4\left(c - \frac{1}{2}\right) = -1 \]

9. **Solving for \( c \)**: Rearranging gives:
\[ 4\left(c - \frac{1}{2}\right) = 1 \implies c - \frac{1}{2} = \frac{1}{4} \implies c = \frac{3}{4} \]

### Final Answer
The value of \( c \) is \( \frac{3}{4} \).