The two ends of latusrectum of a parabola are the points (3, 6) and (-5, 6). The focus, is

Options

A. (1, 6)
B. (-1, 6)
C. (1, -6)
D. (-1, -6)

Answer

To find the focus of the parabola given the endpoints of its latus rectum, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Endpoints of the Latus Rectum**:
The endpoints given are (3, 6) and (-5, 6).

2. **Use the Midpoint Formula**:
The focus of the parabola is located at the midpoint of the endpoints of the latus rectum. The midpoint (M) can be calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the endpoints.

3. **Substitute the Coordinates**:
Here, \((x_1, y_1) = (3, 6)\) and \((x_2, y_2) = (-5, 6)\).
Therefore, we calculate:
\[ M = \left( \frac{3 + (-5)}{2}, \frac{6 + 6}{2} \right) \]

4. **Calculate the x-coordinate of the Midpoint**:
\[ M_x = \frac{3 - 5}{2} = \frac{-2}{2} = -1 \]

5. **Calculate the y-coordinate of the Midpoint**:
\[ M_y = \frac{6 + 6}{2} = \frac{12}{2} = 6 \]

6. **Combine the Coordinates**:
Thus, the coordinates of the focus are:
\[ \text{Focus} = (-1, 6) \]

### Final Answer:
The focus of the parabola is at the point \((-1, 6)\).

---