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

Options

A. `(-7//2, 5//2)`
B. `(-3//4,5//2)`
C. `(-17//4, 5//2)`
D. `(5//2,-3//4)`

Answer

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

### Step 1: Rearrange the equation
We start by rearranging the equation to isolate the terms involving \( y \):

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

### Step 2: Move the \( x \) and constant terms to the other side
Next, we move the \( x \) and constant terms to the right side:

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

### Step 3: Factor out the coefficient of \( y^2 \)
Now, we factor out the 4 from the left side:

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

### Step 4: Complete the square for the \( y \) terms
To complete the square for \( y^2 - 5y \), we take half of the coefficient of \( y \) (which is \(-5\)), square it, and add/subtract it inside the parentheses:

\[ y^2 - 5y = (y - \frac{5}{2})^2 - \left(\frac{5}{2}\right)^2 \]

Calculating \(\left(\frac{5}{2}\right)^2\):

\[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]

So we rewrite the equation:

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

### Step 5: Distribute the 4 and simplify
Distributing the 4 gives:

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

Now, we simplify:

\[ 4(y - \frac{5}{2})^2 = -12x - 67 + 25 \]
\[ 4(y - \frac{5}{2})^2 = -12x - 42 \]

### Step 6: Divide by 4
Now we divide the entire equation by 4:

\[ (y - \frac{5}{2})^2 = -3(x + \frac{21}{2}) \]

### Step 7: Identify the standard form
This equation is now in the standard form of a parabola that opens to the left:

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

where \((h, k)\) is the vertex and \(a\) is the distance from the vertex to the focus.

From our equation, we can identify:

- \(h = -\frac{21}{2}\)
- \(k = \frac{5}{2}\)
- \(4a = 3 \Rightarrow a = \frac{3}{4}\)

### Step 8: Find the focus
The focus of a parabola that opens to the left is located at \((h - a, k)\). Therefore, we calculate:

\[ \text{Focus} = \left(-\frac{21}{2} - \frac{3}{4}, \frac{5}{2}\right) \]

To combine the x-coordinates, we convert \(-\frac{21}{2}\) to a fraction with a denominator of 4:

\[ -\frac{21}{2} = -\frac{42}{4} \]

Now, we can find the focus:

\[ \text{Focus} = \left(-\frac{42}{4} - \frac{3}{4}, \frac{5}{2}\right) = \left(-\frac{45}{4}, \frac{5}{2}\right) \]

### Final Answer
Thus, the focus of the parabola is:

\[ \boxed{\left(-\frac{45}{4}, \frac{5}{2}\right)} \]