The two ends of latus rectum 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 the latus rectum, we follow these steps:

### Step 1: Identify the endpoints of the latus rectum
The endpoints of the latus rectum are given as \( A(3, 6) \) and \( B(-5, 6) \).

### Step 2: Use the midpoint formula to find the focus
The focus of the parabola is located at the midpoint of the latus rectum. The midpoint \( M \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 3: Substitute the coordinates of points A and B into the formula
Here, we have:
- \( x_1 = 3 \), \( y_1 = 6 \)
- \( x_2 = -5 \), \( y_2 = 6 \)

Substituting these values into the midpoint formula:
\[ M = \left( \frac{3 + (-5)}{2}, \frac{6 + 6}{2} \right) \]

### Step 4: Simplify the calculations
Calculating the x-coordinate:
\[ \frac{3 - 5}{2} = \frac{-2}{2} = -1 \]
Calculating the y-coordinate:
\[ \frac{6 + 6}{2} = \frac{12}{2} = 6 \]

### Step 5: Write the coordinates of the focus
Thus, the coordinates of the focus are:
\[ F = (-1, 6) \]

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

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