If the latus rectum of the parabola `2x^(2)- ky + 2 = 0` be 2, then the vertex is

Options

A. `(0, 3/4)`
B. `(0, 1/2)`
C. `(3/4, 0)`
D. `0, 0`

Answer

To find the vertex of the parabola given by the equation \(2x^2 - ky + 2 = 0\) with the condition that the length of the latus rectum is 2, we can follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the parabola:
\[ 2x^2 - ky + 2 = 0 \]
Rearranging gives:
\[ ky = 2x^2 + 2 \]
Dividing through by \(k\) (assuming \(k \neq 0\)):
\[ y = \frac{2}{k}x^2 + \frac{2}{k} \]

### Step 2: Identify the standard form of the parabola
The standard form of a parabola that opens upwards is:
\[ y = ax^2 + bx + c \]
In our case, we can identify:
\[ a = \frac{2}{k}, \quad b = 0, \quad c = \frac{2}{k} \]

### Step 3: Determine the vertex
The vertex of a parabola in the form \(y = ax^2 + bx + c\) is given by the formula:
\[ \left(-\frac{b}{2a}, \frac{4ac - b^2}{4a}\right) \]
Since \(b = 0\), the x-coordinate of the vertex is:
\[ x = 0 \]
Now, substituting \(x = 0\) into the equation to find the y-coordinate:
\[ y = \frac{2}{k}(0)^2 + \frac{2}{k} = \frac{2}{k} \]
Thus, the vertex is:
\[ (0, \frac{2}{k}) \]

### Step 4: Use the latus rectum condition
The length of the latus rectum \(L\) for a parabola is given by:
\[ L = \frac{4}{|a|} \]
Given that \(L = 2\), we set up the equation:
\[ \frac{4}{\left|\frac{2}{k}\right|} = 2 \]
Solving for \(k\):
\[ 4 = 2 \cdot \left|\frac{2}{k}\right| \]
\[ 4 = \frac{4}{|k|} \]
Cross-multiplying gives:
\[ 4|k| = 4 \implies |k| = 1 \]
Thus, \(k = 1\) or \(k = -1\).

### Step 5: Find the vertex for both values of \(k\)
1. If \(k = 1\):
\[ \text{Vertex} = \left(0, \frac{2}{1}\right) = (0, 2) \]

2. If \(k = -1\):
\[ \text{Vertex} = \left(0, \frac{2}{-1}\right) = (0, -2) \]

### Conclusion
The vertices of the parabola are:
- For \(k = 1\): \((0, 2)\)
- For \(k = -1\): \((0, -2)\)

### Final Answer
The vertex of the parabola can be either \((0, 2)\) or \((0, -2)\) depending on the value of \(k\).