Find the equation of the parabola with vertex at the origin and satisying the conditions :
(i) Focus (2,0) , Directrix x = - 2
(ii) Focu (0,-3) , Directirc y = 3 .
(b) Also, find the length of latus in each case.

Answer

To find the equations of the parabolas given the conditions, we will follow these steps:

### Part (i): Focus (2, 0) and Directrix x = -2

1. **Identify the Vertex**:
The vertex of the parabola is at the origin (0, 0).

2. **Determine the value of 'a'**:
The distance from the vertex to the focus is 'a'. The focus is at (2, 0), so:
\[ a = \text{Distance from vertex to focus} = 2 - 0 = 2 \]

3. **Write the equation of the parabola**:
Since the parabola opens towards the right (focus is to the right of the vertex), the standard form of the equation is:
\[ y^2 = 4ax \]
Substituting \(a = 2\):
\[ y^2 = 4 \cdot 2 \cdot x \implies y^2 = 8x \]

4. **Find the length of the latus rectum**:
The length of the latus rectum \(L\) is given by:
\[ L = 4a = 4 \cdot 2 = 8 \]

### Part (ii): Focus (0, -3) and Directrix y = 3

1. **Identify the Vertex**:
The vertex of the parabola is at the origin (0, 0).

2. **Determine the value of 'a'**:
The distance from the vertex to the focus is 'a'. The focus is at (0, -3), so:
\[ a = \text{Distance from vertex to focus} = 0 - (-3) = 3 \]
Since the focus is below the vertex, we take \(a = -3\).

3. **Write the equation of the parabola**:
Since the parabola opens downwards (focus is below the vertex), the standard form of the equation is:
\[ x^2 = 4ay \]
Substituting \(a = -3\):
\[ x^2 = 4 \cdot (-3) \cdot y \implies x^2 = -12y \]

4. **Find the length of the latus rectum**:
The length of the latus rectum \(L\) is given by:
\[ L = 4a = 4 \cdot (-3) = -12 \]
(Note: Length is taken as a positive value, so we consider it as 12).

### Final Answers:
- The equation of the first parabola is \(y^2 = 8x\) with a latus rectum length of 8.
- The equation of the second parabola is \(x^2 = -12y\) with a latus rectum length of 12.