The equation of the latus rectum of the parabola `x^(2)-12x-8y+52=0` is

Options

A. x=4
B. y=4
C. x=6
D. y=2

Answer

To find the equation of the latus rectum of the given parabola \(x^2 - 12x - 8y + 52 = 0\), we will follow these steps:

### Step 1: Rearrange the equation into standard form
We start with the given equation:
\[ x^2 - 12x - 8y + 52 = 0 \]
We can rearrange it to isolate the \(y\) term:
\[ x^2 - 12x + 52 = 8y \]

### Step 2: Complete the square for the \(x\) terms
Next, we complete the square for the \(x\) terms:
\[ x^2 - 12x = (x - 6)^2 - 36 \]
Substituting this back into the equation gives:
\[ (x - 6)^2 - 36 + 52 = 8y \]
Simplifying this:
\[ (x - 6)^2 + 16 = 8y \]
Now, we can express it as:
\[ (x - 6)^2 = 8y - 16 \]

### Step 3: Rearranging to standard parabola form
We can rearrange this to:
\[ (x - 6)^2 = 8(y - 2) \]
This is now in the standard form of a parabola \((x - h)^2 = 4a(y - k)\), where \((h, k)\) is the vertex.

### Step 4: Identify the vertex and the value of \(a\)
From the equation \((x - 6)^2 = 8(y - 2)\), we can see:
- The vertex \((h, k) = (6, 2)\)
- The value \(4a = 8\), hence \(a = 2\)

### Step 5: Write the equation of the latus rectum
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry that passes through the focus. For a parabola that opens upwards, the equation of the latus rectum is given by \(y = k\), where \(k\) is the y-coordinate of the vertex.

In our case, since the vertex is at \((6, 2)\), the equation of the latus rectum is:
\[ y = 2 \]

### Conclusion
Thus, the equation of the latus rectum of the parabola \(x^2 - 12x - 8y + 52 = 0\) is:
\[ \boxed{y = 2} \]