Find th equations of axis and directrix of the parabola
`4x^(2)` +12x-20y+67=0

Answer

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

### Step 1: Rearrange the equation
Start by rearranging the given equation to isolate the \(y\) terms on one side.

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

### Step 2: Divide by 4
Next, divide the entire equation by 4 to simplify it.

\[ x^2 + 3x + \frac{67}{4} = 5y \]

### Step 3: Complete the square
To complete the square for the \(x\) terms, take the coefficient of \(x\) (which is 3), halve it to get \(\frac{3}{2}\), and then square it to get \(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\). Add and subtract this value inside the equation.

\[ x^2 + 3x + \frac{9}{4} - \frac{9}{4} + \frac{67}{4} = 5y \]

This simplifies to:

\[ \left(x + \frac{3}{2}\right)^2 - \frac{9}{4} + \frac{67}{4} = 5y \]

Combine the constants:

\[ \left(x + \frac{3}{2}\right)^2 + \frac{58}{4} = 5y \]

### Step 4: Rearrange to standard form
Rearranging gives:

\[ \left(x + \frac{3}{2}\right)^2 = 5y - \frac{58}{4} \]

This can be rewritten as:

\[ \left(x + \frac{3}{2}\right)^2 = 5\left(y - \frac{29}{10}\right) \]

### Step 5: Identify parameters
Now, we can identify the parameters from the standard form of the parabola \((x - h)^2 = 4a(y - k)\):

- \(h = -\frac{3}{2}\)
- \(k = \frac{29}{10}\)
- \(4a = 5\) which gives \(a = \frac{5}{4}\)

### Step 6: Find the axis of the parabola
The axis of the parabola is given by the line \(x = h\). Therefore, substituting for \(h\):

\[ x + \frac{3}{2} = 0 \quad \text{or} \quad 2x + 3 = 0 \]

### Step 7: Find the directrix
The directrix is given by the equation \(y = k - a\). Substituting the values of \(k\) and \(a\):

\[ y = \frac{29}{10} - \frac{5}{4} \]

To subtract these fractions, convert \(\frac{5}{4}\) to a fraction with a denominator of 20:

\[ \frac{5}{4} = \frac{25}{20} \]

Thus:

\[ y = \frac{29}{10} - \frac{25}{20} = \frac{58}{20} - \frac{25}{20} = \frac{33}{20} \]

### Final Results
- The equation of the axis is \(2x + 3 = 0\).
- The equation of the directrix is \(y = \frac{33}{20}\).