If `x=my+c` is a normal to the parabola `x^(2)=4ay`, then the value of c, is

Options

A. `-2am-am^(3)`
B. `2am+am^(3)`
C. `-(2a)/m-a/m^(3)`
D. `(2a)/m+a/m^(3)`

Answer

To solve the problem of finding the value of \( c \) such that the line \( x = my + c \) is a normal to the parabola \( x^2 = 4ay \), we can follow these steps:

### Step 1: Understand the given parabola and the normal equation
The given parabola is \( x^2 = 4ay \). The general equation of the normal to this parabola in slope form is given by:
\[ y = mx + 2a - \frac{a}{m^2} \]
where \( m \) is the slope of the normal line.

### Step 2: Rewrite the normal line equation
The normal line is given as \( x = my + c \). We can rearrange this to express \( y \) in terms of \( x \):
\[ y = \frac{x - c}{m} \]

### Step 3: Compare the two equations
Now we have two equations:
1. The normal line: \( y = \frac{x - c}{m} \)
2. The normal to the parabola: \( y = mx + 2a - \frac{a}{m^2} \)

We will compare the coefficients of \( x \) and the constant terms from both equations.

### Step 4: Coefficient of \( x \)
From the normal line, the coefficient of \( x \) is \( \frac{1}{m} \), and from the normal equation of the parabola, the coefficient of \( x \) is \( m \). Thus, we set them equal:
\[ \frac{1}{m} = m \]
Multiplying both sides by \( m \) (assuming \( m \neq 0 \)):
\[ 1 = m^2 \implies m = 1 \text{ or } m = -1 \]

### Step 5: Constant terms comparison
Now we compare the constant terms. From the normal line, the constant term is \( -\frac{c}{m} \), and from the normal equation of the parabola, the constant term is \( 2a - \frac{a}{m^2} \). Thus, we have:
\[ -\frac{c}{m} = 2a - \frac{a}{m^2} \]

### Step 6: Substitute the value of \( m \)
Let's first consider \( m = 1 \):
\[ -\frac{c}{1} = 2a - a \implies -c = a \implies c = -a \]

Now consider \( m = -1 \):
\[ -\frac{c}{-1} = 2a - \frac{a}{(-1)^2} \implies c = 2a - a \implies c = a \]

### Conclusion
Thus, we have two possible values for \( c \):
1. If \( m = 1 \), then \( c = -a \).
2. If \( m = -1 \), then \( c = a \).

### Final Answer
The value of \( c \) can be either \( -a \) or \( a \).

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