If the chord `y = mx + c` subtends a right angle at the vertex of the parabola `y^2 = 4ax`, thenthe value of c is

Options

A. `-4 am`
B. `4 am`
C. `-2 am`
D. `2 am`

Answer

To solve the problem, we need to find the value of \( c \) such that the chord \( y = mx + c \) subtends a right angle at the vertex of the parabola \( y^2 = 4ax \).

### Step-by-Step Solution:

1. **Understand the Parabola**:
The given parabola is \( y^2 = 4ax \). The vertex of this parabola is at the origin \( (0,0) \).

2. **Equation of the Chord**:
The equation of the chord is given as \( y = mx + c \).

3. **Homogenization**:
We can express the equation of the chord in a different form to facilitate our calculations. We rearrange it as:
\[ y - mx = c \]
This can be rewritten as:
\[ \frac{y - mx}{c} = 1 \]

4. **Substituting into the Parabola**:
We can replace \( 1 \) in the parabola's equation \( y^2 = 4ax \) with \( \frac{y - mx}{c} \):
\[ y^2 = 4ax \cdot \frac{y - mx}{c} \]
Rearranging gives:
\[ cy^2 = 4ax(y - mx) \]

5. **Expanding the Equation**:
Expanding the right-hand side:
\[ cy^2 = 4axy - 4amxy \]
Rearranging this gives:
\[ 4amx^2 - 4axy + cy^2 = 0 \]

6. **Identifying the Coefficients**:
The equation \( 4amx^2 - 4axy + cy^2 = 0 \) represents a pair of straight lines (the lines OP and OQ). For these lines to be perpendicular, the condition is:
\[ \text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0 \]
This gives us:
\[ 4am + c = 0 \]

7. **Solving for \( c \)**:
From the equation \( 4am + c = 0 \), we can solve for \( c \):
\[ c = -4am \]

### Final Answer:
Thus, the value of \( c \) is:
\[ \boxed{-4am} \]