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

Options

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

Answer

To find the vertex 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 group the \(y\) terms together:

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

### Step 2: Move the \(x\) terms to the other side
Next, we isolate the \(y\) terms on one 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 the expression \(y^2 - 5y\), we take half of the coefficient of \(y\) (which is \(-5\)), square it, and add/subtract it inside the parentheses:

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

This gives us:

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

### Step 5: Substitute back into the equation
Substituting this back into our equation gives:

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

Distributing the 4:

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

### Step 6: Move the constant to the other side
Now, we move \(-25\) to the right side:

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

This simplifies to:

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

### Step 7: Divide by 4
To simplify further, we divide the entire equation by 4:

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

### Step 8: Rewrite in vertex form
We can rewrite this equation as:

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

### Step 9: Identify the vertex
Now, we can identify the vertex from the equation in the form \((y - k)^2 = -4a(x - h)\). Here, \(h = -\frac{7}{2}\) and \(k = \frac{5}{2}\).

Thus, the vertex of the parabola is:

\[ \left(-\frac{7}{2}, \frac{5}{2}\right) \]

### Final Answer
The vertex of the parabola \(4y^2 + 12x - 20y + 67 = 0\) is \(\left(-\frac{7}{2}, \frac{5}{2}\right)\).

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