If the vertex of a parabola is (0, 2) and the extremities of latusrectum are (-6, 4) and (6, 4) then, its equation, is

Options

A. `x^(2)-4y+8=0`
B. `x^(2)+4y-8=0`
C. `x^(2)-8y+16=0`
D. `x^(2)+8y-16=0`

Answer

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

### Step 1: Identify the Vertex and Extremities of the Latus Rectum
- The vertex of the parabola is given as \( (0, 2) \).
- The extremities of the latus rectum are given as \( (-6, 4) \) and \( (6, 4) \).

### Step 2: Find the Focus
- The focus of the parabola is the midpoint of the extremities of the latus rectum.
- To find the midpoint, we use the formula:
\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- Here, \( (x_1, y_1) = (-6, 4) \) and \( (x_2, y_2) = (6, 4) \).
- Calculating the midpoint:
\[ \text{Focus} = \left( \frac{-6 + 6}{2}, \frac{4 + 4}{2} \right) = \left( 0, 4 \right) \]

### Step 3: Determine the Orientation of the Parabola
- The vertex is at \( (0, 2) \) and the focus is at \( (0, 4) \).
- Since the focus is above the vertex, the parabola opens upwards.

### Step 4: Calculate the Distance \( A \)
- The distance \( A \) between the vertex and the focus is:
\[ A = \text{Focus y-coordinate} - \text{Vertex y-coordinate} = 4 - 2 = 2 \]

### Step 5: Use the Standard Form of the Parabola
- The standard form of a parabola that opens upwards is given by:
\[ (x - h)^2 = 4A(y - k) \]
- Here, \( (h, k) \) is the vertex, which is \( (0, 2) \), and \( A = 2 \).

### Step 6: Substitute Values into the Equation
- Substituting \( h = 0 \), \( k = 2 \), and \( A = 2 \):
\[ (x - 0)^2 = 4 \cdot 2 \cdot (y - 2) \]
\[ x^2 = 8(y - 2) \]

### Step 7: Rearranging the Equation
- Expanding the equation:
\[ x^2 = 8y - 16 \]
- Rearranging gives:
\[ x^2 - 8y + 16 = 0 \]

### Final Answer
The equation of the parabola is:
\[ x^2 - 8y + 16 = 0 \]

---