The length of latus rectum `x^(2) - 6x+ 16y + 25 = 0` is

Options

A. a) 5
B. b) 4
C. c) 16
D. d) `(5)/(3)`

Answer

To find the length of the latus rectum of the parabola given by the equation \( x^2 - 6x + 16y + 25 = 0 \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ x^2 - 6x + 16y + 25 = 0 \]
We can rearrange it to isolate \( y \):
\[ 16y = -x^2 + 6x - 25 \]
\[ y = -\frac{1}{16}(x^2 - 6x + 25) \]

### Step 2: Complete the square for the \( x \) terms
Next, we complete the square for the quadratic expression in \( x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
Substituting this back into the equation gives:
\[ y = -\frac{1}{16}((x - 3)^2 - 9 + 25) \]
\[ y = -\frac{1}{16}((x - 3)^2 + 16) \]
\[ y = -\frac{1}{16}(x - 3)^2 - 1 \]

### Step 3: Rewrite in standard form
Now, we can rewrite the equation in the standard form of a parabola:
\[ (x - 3)^2 = -16(y + 1) \]
This is in the form \( (x - h)^2 = -4p(y - k) \), where \( (h, k) \) is the vertex of the parabola.

### Step 4: Identify \( p \)
From the standard form, we can see that \( 4p = 16 \). Thus, we find \( p \):
\[ p = \frac{16}{4} = 4 \]

### Step 5: Calculate 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 \( p \):
\[ L = 4 \times 4 = 16 \]

### Final Answer
The length of the latus rectum is \( 16 \).

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