The length of the latus - rectum of the parabola `4y^(2) + 2x - 20 y + 17 = 0 ` is

Options

A. 2
B. `(1)/(8)`
C. `(1)/(2)`
D. 4

Answer

To find the length of the latus rectum of the parabola given by the equation \(4y^2 + 2x - 20y + 17 = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
First, we will rearrange the equation to isolate the \(x\) terms on one side and the \(y\) terms on the other side.

\[ 4y^2 - 20y + 2x + 17 = 0 \implies 2x = -4y^2 + 20y - 17 \]

### Step 2: Completing the Square
Next, we will complete the square for the \(y\) terms. We take the \(4y^2 - 20y\) part:

1. Factor out the coefficient of \(y^2\):
\[ 4(y^2 - 5y) \]

2. To complete the square, take half of the coefficient of \(y\) (which is \(-5\)), square it, and add and subtract it inside the parentheses:
\[ = 4\left(y^2 - 5y + \left(\frac{5}{2}\right)^2 - \left(\frac{5}{2}\right)^2\right) \]
\[ = 4\left((y - \frac{5}{2})^2 - \frac{25}{4}\right) \]
\[ = 4(y - \frac{5}{2})^2 - 25 \]

3. Substitute back into the equation:
\[ 2x = -4(y - \frac{5}{2})^2 + 25 - 17 \]
\[ 2x = -4(y - \frac{5}{2})^2 + 8 \]

### Step 3: Rearranging to Standard Form
Now, we rearrange the equation to the standard form of a parabola:
\[ 4(y - \frac{5}{2})^2 = -2(x - 4) \]

This can be rewritten as:
\[ (y - \frac{5}{2})^2 = -\frac{1}{2}(x + 4) \]

### Step 4: Identifying Parameters
From the standard form \((y - k)^2 = 4p(x - h)\), we can identify:
- \(k = \frac{5}{2}\)
- \(h = -4\)
- \(4p = -\frac{1}{2}\) which gives \(p = -\frac{1}{8}\)

### Step 5: Finding the Length of the Latus Rectum
The length of the latus rectum \(L\) of a parabola is given by the formula:
\[ L = |4p| \]
Substituting the value of \(p\):
\[ L = |4 \times -\frac{1}{8}| = |-\frac{4}{8}| = \frac{1}{2} \]

### Final Answer
The length of the latus rectum of the parabola is \(\frac{1}{2}\) units.

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