The vertex of a parabola is the point (a,b) and latus rectum is of length I. If the axis of the parabola is along the positive direction of y-axis, then its equation is

Options

A. `(x+a)^2=l/2(2y-2b)`
B. `(x-a)^2=l/2(2y-2b)`
C. `(x+a)^2=l/4(2y-2b)`
D. `(x-a)^2=l/8(2y-2b)`

Answer

To find the equation of the parabola given the vertex (a, b) and the length of the latus rectum L, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vertex and Latus Rectum**:
- The vertex of the parabola is given as (a, b).
- The length of the latus rectum is given as L.

2. **Understand the Orientation of the Parabola**:
- Since the axis of the parabola is along the positive direction of the y-axis, the general form of the equation of the parabola is:
\[ (x - h)^2 = 4p(y - k) \]
where (h, k) is the vertex.

3. **Substituting the Vertex**:
- Here, h = a and k = b. Therefore, we can substitute these values into the equation:
\[ (x - a)^2 = 4p(y - b) \]

4. **Relate Latus Rectum to p**:
- The length of the latus rectum (L) is given by the formula \( L = 4p \).
- From this, we can express p in terms of L:
\[ p = \frac{L}{4} \]

5. **Substituting p into the Equation**:
- Now, substitute \( p = \frac{L}{4} \) back into the parabola equation:
\[ (x - a)^2 = 4\left(\frac{L}{4}\right)(y - b) \]
- Simplifying this gives:
\[ (x - a)^2 = L(y - b) \]

6. **Final Equation of the Parabola**:
- Thus, the equation of the parabola is:
\[ (x - a)^2 = L(y - b) \]