The length of the latusrectum of the parabola `x=ay^(2)+by+c,` is

Options

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

Explanation

To find the length of the latus rectum of the parabola given by the equation \( x = ay^2 + by + c \), we can follow these steps:

### Step 1: Rewrite the equation
Start with the equation of the parabola:
\[ x = ay^2 + by + c \]
To make it easier to work with, divide the entire equation by \( a \):
\[ \frac{x}{a} = y^2 + \frac{b}{a}y + \frac{c}{a} \]

### Step 2: Complete the square
Next, we need to complete the square for the \( y \) terms. The expression \( y^2 + \frac{b}{a}y \) can be rewritten by completing the square:
\[ y^2 + \frac{b}{a}y = \left(y + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 \]
Substituting this back into the equation gives:
\[ \frac{x}{a} = \left(y + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2 + \frac{c}{a} \]

### Step 3: Rearrange the equation
Rearranging the equation, we have:
\[ \frac{x}{a} + \left(\frac{b}{2a}\right)^2 - \frac{c}{a} = \left(y + \frac{b}{2a}\right)^2 \]
Multiplying through by \( a \) leads to:
\[ x + \frac{b^2}{4a} - c = a\left(y + \frac{b}{2a}\right)^2 \]

### Step 4: Identify the standard form
This can be expressed in the standard form of a parabola:
\[ y - k = \frac{1}{4p}(x - h) \]
where \( p \) is the distance from the vertex to the focus, and \( 4p \) is the length of the latus rectum.

### Step 5: Determine the coefficient
From our rearranged equation, we can see that:
\[ 4p = \frac{1}{a} \]
Thus, the length of the latus rectum \( L \) is:
\[ L = 4p = \frac{1}{a} \]

### Conclusion
The length of the latus rectum of the parabola \( x = ay^2 + by + c \) is:
\[ \frac{1}{a} \]