If three normals are drawn from the point (c, 0) to the parabola `y^(2)=4x` and two of which are perpendicular, then the value of c is equal to

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to find the value of \( c \) such that three normals drawn from the point \( (c, 0) \) to the parabola \( y^2 = 4x \) include two that are perpendicular to each other.

### Step 1: Understand the parabola and its properties
The given parabola is \( y^2 = 4x \). This can be rewritten in the standard form \( (y - 0)^2 = 4(1)(x - 0) \), where \( a = 1 \). The vertex of the parabola is at the origin \( (0, 0) \).

### Step 2: Determine the slopes of the normals
For a point \( P(h, k) \) on the parabola, the slope of the normal can be derived from the slope of the tangent at that point. The slope of the tangent at \( P(h, k) \) is given by:
\[ m_t = -\frac{y}{2} \Bigg|_{(h, k)} = -\frac{k}{2} \]
Thus, the slope of the normal \( m \) is:
\[ m = \frac{2}{k} \]

### Step 3: Find the slopes of the normals from the point \( (c, 0) \)
The normals from the point \( (c, 0) \) to the parabola will have slopes \( m_1, m_2, m_3 \). Since we have three normals, we can express their slopes in terms of the coordinates of the points on the parabola.

### Step 4: Use the property of co-normal points
For three normals drawn from a point, if two of them are perpendicular, then:
\[ m_1 m_2 = -1 \]
Additionally, we have the following relationships for co-normal points:
1. \( m_1 + m_2 + m_3 = 0 \)
2. \( m_1 m_2 + m_2 m_3 + m_3 m_1 = 2a - h \)
3. \( m_1 m_2 m_3 = -k \)

### Step 5: Substitute known values
From the problem, we know:
- \( h = c \)
- \( k = 0 \)
- \( a = 1 \)

Thus, we can rewrite the equations:
1. \( m_1 + m_2 + m_3 = 0 \)
2. \( m_1 m_2 + m_2 m_3 + m_3 m_1 = 2(1) - c = 2 - c \)
3. \( m_1 m_2 m_3 = 0 \)

### Step 6: Analyze the equations
Since \( m_1 m_2 = -1 \), we can denote \( m_1 = m \) and \( m_2 = -\frac{1}{m} \). Then, substituting into the first equation:
\[ m - \frac{1}{m} + m_3 = 0 \implies m_3 = -\left(m - \frac{1}{m}\right) = -m + \frac{1}{m} \]

### Step 7: Substitute into the second equation
Now substituting \( m_1 \) and \( m_2 \) into the second equation:
\[ m \left(-\frac{1}{m}\right) + \left(-\frac{1}{m}\right)m_3 + m m_3 = 2 - c \]
This simplifies to:
\[ -1 - \frac{1}{m} m_3 + m m_3 = 2 - c \]

### Step 8: Solve for \( c \)
Substituting \( m_3 = -m + \frac{1}{m} \) into the equation, we can solve for \( c \). After simplification, we will find:
\[ c = 3 \]

### Final Answer
Thus, the value of \( c \) is \( 3 \).