The equation of the parabola whose vertex is `(2,0)` and extremities of latus rectum are `(3, 2)` and `(3,-2)` is

Options

A. `y^2 = 2x– 4`
B. `y^2 = 4x-8`
C. `x^2 = 4y - 3`
D. none

Answer

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

### Step 1: Identify the vertex and the extremities of the latus rectum
The vertex of the parabola is given as \( (2, 0) \) and the extremities of the latus rectum are \( (3, 2) \) and \( (3, -2) \).

### Step 2: Determine the orientation of the parabola
Since the x-coordinates of the extremities of the latus rectum are the same (both are \( x = 3 \)), the parabola opens horizontally. The general form of a horizontally opening parabola is:
\[ (y - k)^2 = 4a(x - h) \]
where \( (h, k) \) is the vertex.

### Step 3: Substitute the vertex into the equation
Substituting the vertex \( (h, k) = (2, 0) \) into the equation gives:
\[ (y - 0)^2 = 4a(x - 2) \]
which simplifies to:
\[ y^2 = 4a(x - 2) \]

### Step 4: Calculate the distance of the latus rectum
The length of the latus rectum is given by the distance between the extremities of the latus rectum. The distance between the points \( (3, 2) \) and \( (3, -2) \) can be calculated as:
\[ \text{Distance} = |y_2 - y_1| = |2 - (-2)| = |2 + 2| = 4 \]
The length of the latus rectum is also given by \( 4a \). Therefore, we have:
\[ 4a = 4 \]

### Step 5: Solve for \( a \)
Dividing both sides by 4 gives:
\[ a = 1 \]

### Step 6: Substitute \( a \) back into the equation
Now substituting \( a = 1 \) back into the equation:
\[ y^2 = 4(1)(x - 2) \]
This simplifies to:
\[ y^2 = 4(x - 2) \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ y^2 = 4x - 8 \]

### Final Answer
Thus, the equation of the parabola is:
\[ y^2 = 4x - 8 \]