The vertex of a parabola is (2, 2) and the coordinats of its two extremities of latus rectum are `(-2,0)` and (6, 0). Then find the equation of the parabola.

Explanation

To find the equation of the parabola given the vertex and the coordinates of the extremities of the latus rectum, we can follow these steps:

### Step 1: Identify the vertex and the coordinates of the latus rectum
The vertex of the parabola is given as \( (h, k) = (2, 2) \). The coordinates of the extremities of the latus rectum are given as \( (-2, 0) \) and \( (6, 0) \).

### Step 2: Find the focus of the parabola
The focus of the parabola lies on the line segment joining the two extremities of the latus rectum. To find the focus, we first calculate the midpoint of the extremities:

\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-2 + 6}{2}, \frac{0 + 0}{2} \right) = \left( \frac{4}{2}, 0 \right) = (2, 0) \]

### Step 3: Determine the distance \( a \)
The distance \( a \) is the distance from the vertex to the focus. Since the vertex is at \( (2, 2) \) and the focus is at \( (2, 0) \), we can calculate \( a \):

\[ a = |k - y_{\text{focus}}| = |2 - 0| = 2 \]

### Step 4: Determine the orientation of the parabola
Since the focus is below the vertex, the parabola opens downwards.

### Step 5: Write the equation of the parabola
The standard form of the equation of a parabola that opens downwards is:

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

Substituting \( h = 2 \), \( k = 2 \), and \( a = 2 \):

\[ (x - 2)^2 = -4 \cdot 2 (y - 2) \]

This simplifies to:

\[ (x - 2)^2 = -8(y - 2) \]

### Final Equation
Thus, the equation of the parabola is:

\[ (x - 2)^2 = -8(y - 2) \]