Find the vertex , focus, axis , directrix and latusrectum of the parabola `4y^2+12x-20y+67=0` .

Answer

To solve the problem of finding the vertex, focus, axis, directrix, and latus rectum of the parabola given by the equation \(4y^2 + 12x - 20y + 67 = 0\), we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to isolate the quadratic terms. We will move the linear terms to the right side.

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

### Step 2: Completing the Square
Next, we will complete the square for the \(y\) terms. Factor out the coefficient of \(y^2\) (which is 4) from the left side:

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

Now, 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. Half of \(-5\) is \(-\frac{5}{2}\), and squaring it gives \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\).

So we rewrite the equation:

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

This simplifies to:

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

Distributing the 4 gives:

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

Adding 25 to both sides:

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

### Step 3: Simplifying the Equation
Now, we can divide the entire equation by 4 to simplify it further:

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

Rearranging gives:

\[ (y - \frac{5}{2})^2 = -3\left(x + \frac{21}{6}\right) \]

### Step 4: Identifying Parameters
This equation is now in the form \((y - k)^2 = -4a(x - h)\), where \(k = \frac{5}{2}\), \(h = -\frac{21}{6}\), and \(4a = 3\), thus \(a = \frac{3}{4}\).

### Step 5: Finding the Vertex
The vertex \((h, k)\) is:

\[ \text{Vertex} = \left(-\frac{21}{6}, \frac{5}{2}\right) = \left(-\frac{7}{2}, \frac{5}{2}\right) \]

### Step 6: Finding the Focus
The focus is located at \((h - a, k)\):

\[ \text{Focus} = \left(-\frac{7}{2} - \frac{3}{4}, \frac{5}{2}\right) = \left(-\frac{14}{4} - \frac{3}{4}, \frac{5}{2}\right) = \left(-\frac{17}{4}, \frac{5}{2}\right) \]

### Step 7: Finding the Axis
The axis of the parabola is the line \(y = k\):

\[ \text{Axis} = y = \frac{5}{2} \]

### Step 8: Finding the Directrix
The directrix is given by the equation \(x = h + a\):

\[ \text{Directrix} = x = -\frac{7}{2} + \frac{3}{4} = -\frac{14}{4} + \frac{3}{4} = -\frac{11}{4} \]

### Step 9: Finding the Length of the Latus Rectum
The length of the latus rectum is \(4a\):

\[ \text{Length of Latus Rectum} = 4 \times \frac{3}{4} = 3 \]

### Summary of Results
- **Vertex**: \(\left(-\frac{7}{2}, \frac{5}{2}\right)\)
- **Focus**: \(\left(-\frac{17}{4}, \frac{5}{2}\right)\)
- **Axis**: \(y = \frac{5}{2}\)
- **Directrix**: \(x = -\frac{11}{4}\)
- **Length of Latus Rectum**: \(3\)