The ends of latusrectum of parabola are (4, 8) and (4,-8) then equation of parabola is

Options

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

Answer

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

### Step 1: Identify the coordinates of the ends of the latus rectum
The ends of the latus rectum are given as (4, 8) and (4, -8).

### Step 2: Determine the focus of the parabola
The focus of the parabola lies at the midpoint of the latus rectum. To find the midpoint, we can use the formula for the midpoint of two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of the ends of the latus rectum:
\[ \text{Midpoint} = \left(\frac{4 + 4}{2}, \frac{8 + (-8)}{2}\right) = \left(4, 0\right) \]
Thus, the focus of the parabola is at \( (4, 0) \).

### Step 3: Determine the length of the latus rectum
The length of the latus rectum is the distance between the two points on the y-axis. The distance can be calculated as:
\[ \text{Length of latus rectum} = |y_1 - y_2| = |8 - (-8)| = 8 + 8 = 16 \]

### Step 4: Relate the length of the latus rectum to \(a\)
The length of the latus rectum is given by the formula \(4a\). Therefore, we can set up the equation:
\[ 4a = 16 \]
Solving for \(a\):
\[ a = \frac{16}{4} = 4 \]

### Step 5: Write the equation of the parabola
Since the parabola opens to the right and is symmetric about the x-axis, its standard equation is:
\[ y^2 = 4ax \]
Substituting \(a = 4\):
\[ y^2 = 4 \cdot 4 \cdot x = 16x \]

### Final Answer
Thus, the equation of the parabola is:
\[ \boxed{y^2 = 16x} \]