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 can be drawn to the curve \( y^2 = x \) from the point \( (c, 0) \), with one normal being the x-axis and the other two being perpendicular to each other.

### Step-by-Step Solution:

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

2. **Normal equation for the parabola**: The equation of the normal to the parabola \( y^2 = 4ax \) at a point \( (at^2, 2at) \) is given by:
\[ y = mx - 2a(m - t) \]
For our curve \( y^2 = x \), we have \( 4a = 1 \) which gives \( a = \frac{1}{4} \).

3. **Equation of the normal**: The normal can be expressed as:
\[ y = mx - \frac{1}{2}(m - t) \]
where \( m \) is the slope of the normal and \( t \) is the parameter corresponding to the point on the parabola.

4. **Substituting the point \( (c, 0) \)**: Since the normal passes through \( (c, 0) \):
\[ 0 = mc - \frac{1}{2}(m - t) \]
Rearranging gives:
\[ mc = \frac{1}{2}(m - t) \]
Multiplying through by 2:
\[ 2mc = m - t \]
Thus, we can express \( t \) as:
\[ t = m - 2mc \]

5. **Finding the relationship between \( m \) and \( c \)**: The point on the parabola corresponding to \( t \) is \( (c, 0) \) and must satisfy \( y^2 = x \). Therefore:
\[ (m - 2mc)^2 = 4a(c) \]
Substituting \( a = \frac{1}{4} \):
\[ (m - 2mc)^2 = c \]

6. **Finding the slopes**: We know that one of the normals is the x-axis, which corresponds to \( m = 0 \). The other two normals must satisfy the condition that they are perpendicular:
\[ m_1 m_2 = -1 \]

7. **Using the perpendicular condition**: Let \( m_1 = 2\sqrt{c - \frac{1}{2}} \) and \( m_2 = -2\sqrt{c - \frac{1}{2}} \). Then:
\[ (2\sqrt{c - \frac{1}{2}})(-2\sqrt{c - \frac{1}{2}}) = -1 \]
This simplifies to:
\[ -4(c - \frac{1}{2}) = -1 \]
Thus:
\[ 4(c - \frac{1}{2}) = 1 \]
Rearranging gives:
\[ c - \frac{1}{2} = \frac{1}{4} \]
Therefore:
\[ c = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \]

8. **Conclusion**: The value of \( c \) is:
\[ c = \frac{3}{4} \]