The length of the latus rectum of the parabola `4y^(2)+12x-20y+67=0` is

Options

A. 2
B. 1
C. 8
D. 3

Answer

To find the length of the latus rectum of the parabola given by the equation \(4y^2 + 12x - 20y + 67 = 0\), we will follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation to isolate the quadratic terms.

\[ 4y^2 - 20y + 12x + 67 = 0 \]

### Step 2: Group the \(y\) terms
Group the \(y\) terms together:

\[ 4y^2 - 20y = -12x - 67 \]

### Step 3: Complete the square for \(y\)
To complete the square for the \(y\) terms, factor out the coefficient of \(y^2\):

\[ 4(y^2 - 5y) = -12x - 67 \]

Now, complete the square inside the parentheses. Take half of the coefficient of \(y\) (which is -5), square it, and add it inside the parentheses:

\[ 4\left(y^2 - 5y + \frac{25}{4} - \frac{25}{4}\right) = -12x - 67 \]

This simplifies to:

\[ 4\left((y - \frac{5}{2})^2 - \frac{25}{4}\right) = -12x - 67 \]

Distributing the 4 gives:

\[ 4(y - \frac{5}{2})^2 - 25 = -12x - 67 \]

### Step 4: Move constants to the right side
Now, move the constant to the right side:

\[ 4(y - \frac{5}{2})^2 = -12x - 67 + 25 \]

This simplifies to:

\[ 4(y - \frac{5}{2})^2 = -12x - 42 \]

### Step 5: Divide by 4
Divide the entire equation by 4 to simplify:

\[ (y - \frac{5}{2})^2 = -3(x + \frac{7}{2}) \]

### Step 6: Identify the standard form
The equation is now in the standard form of a parabola that opens to the left:

\[ (y - k)^2 = -4p(x - h) \]

Here, \(k = \frac{5}{2}\), \(h = -\frac{7}{2}\), and \(4p = 3\).

### Step 7: Calculate \(p\)
From \(4p = 3\), we can find \(p\):

\[ p = \frac{3}{4} \]

### Step 8: Length of the latus rectum
The length of the latus rectum of a parabola is given by \(4p\). Therefore, we can calculate:

\[ \text{Length of latus rectum} = 4p = 4 \times \frac{3}{4} = 3 \]

### Final Answer
Thus, the length of the latus rectum of the parabola is \(3\).

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