Equation of parabola ,having (4,-8) and (4,8) as ends of its latus-rectum , is

Options

A. `y^(2) = 8x`
B. `y^(2) = 16x `
C. `y^(2) = - 32x `
D. `x^(2) = 16y`

Answer

To find the equation of the parabola that has the points (4, -8) and (4, 8) as the ends of its latus rectum, we can follow these steps:

### Step 1: Identify the properties of the parabola
The points (4, -8) and (4, 8) indicate that the latus rectum is vertical, meaning the parabola opens either upwards or downwards. The x-coordinate of the focus will be the same as the x-coordinates of the endpoints of the latus rectum.

### Step 2: Find the focus and directrix
The midpoint of the latus rectum will give us the focus of the parabola. We can calculate the midpoint of the points (4, -8) and (4, 8):

\[ \text{Midpoint} = \left( \frac{4 + 4}{2}, \frac{-8 + 8}{2} \right) = (4, 0) \]

So, the focus of the parabola is at (4, 0).

### Step 3: Calculate the distance 'a'
The distance from the focus to either endpoint of the latus rectum gives us 'a'. The distance from (4, 0) to (4, 8) is:

\[ \text{Distance} = |8 - 0| = 8 \]

Since the latus rectum is 2a, we have:

\[ 2a = 8 \implies a = 4 \]

### Step 4: Write the equation of the parabola
For a parabola that opens upwards with the vertex at the origin (0, 0) and focus at (4, 0), the standard form of the equation is:

\[ y^2 = 4ax \]

Substituting \(a = 4\):

\[ y^2 = 4 \cdot 4 \cdot x \implies y^2 = 16x \]

### Final Answer
The equation of the parabola is:

\[ y^2 = 16x \]

---