The vertex of a parabola is the point (a,b) and latusrectum is of length `l`. 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(2 y-2b)`
B. `(x-a)^(2)=l/2(2 y-2b)`
C. `(x+a)^(2)=l/4(2 y-2b)`
D. `(x-a)^(2)=l/8(2 y-2b)`

Explanation

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

### Step 1: Understand the standard form of the parabola
Since the axis of the parabola is along the positive direction of the y-axis, the standard form of the equation of the parabola can be expressed as:
\[ x^2 = 4Ay \]
Here, \(A\) is a constant that determines the distance from the vertex to the focus.

### Step 2: Shift the parabola to the vertex (a, b)
To shift the vertex from the origin (0, 0) to the point (a, b), we replace \(x\) with \(x - a\) and \(y\) with \(y - b\). Thus, the equation becomes:
\[ (x - a)^2 = 4A(y - b) \]

### Step 3: Relate the latus rectum to the constant A
The length of the latus rectum \(L\) of a parabola is given by the formula:
\[ L = 4A \]
From this, we can express \(A\) in terms of \(L\):
\[ A = \frac{L}{4} \]

### Step 4: Substitute A into the equation
Now, substituting \(A = \frac{L}{4}\) into the equation we derived in Step 2:
\[ (x - a)^2 = 4\left(\frac{L}{4}\right)(y - b) \]
This simplifies to:
\[ (x - a)^2 = L(y - b) \]

### Final Equation
Thus, the final equation of the parabola with vertex (a, b) and latus rectum of length \(L\) is:
\[ (x - a)^2 = L(y - b) \]