If the focus of the parabola `x^2-k y+3=0` is (0,2), then a values of `k` is (are) 4 (b) 6 (c) 3 (d) 2

Options

A. 4
B. 6
C. 3
D. 2

Explanation

To solve the problem, we need to find the value(s) of \( k \) such that the focus of the parabola given by the equation \( x^2 - ky + 3 = 0 \) is at the point \( (0, 2) \).

### Step-by-Step Solution:

1. **Rewrite the Parabola Equation**:
Start by rearranging the given equation \( x^2 - ky + 3 = 0 \) to isolate \( y \):
\[ x^2 = ky - 3 \]
\[ ky = x^2 + 3 \]
\[ y = \frac{x^2 + 3}{k} \]

2. **Identify the Standard Form**:
The standard form of a parabola that opens upwards is given by:
\[ (x - h)^2 = 4a(y - k) \]
From our rearranged equation, we can see that:
\[ (x - 0)^2 = k\left(y - \frac{3}{k}\right) \]
Here, \( h = 0 \) and \( k = \frac{3}{k} \).

3. **Determine the Focus**:
For a parabola in the form \( (x - h)^2 = 4a(y - k) \), the focus is located at \( (h, k + a) \).
From our equation, we have:
\[ a = \frac{k}{4} \]
Therefore, the focus is at:
\[ (0, \frac{3}{k} + \frac{k}{4}) \]

4. **Set the Focus Equal to (0, 2)**:
Since we know the focus is at \( (0, 2) \), we can set up the equation:
\[ \frac{3}{k} + \frac{k}{4} = 2 \]

5. **Clear the Fractions**:
Multiply through by \( 4k \) to eliminate the denominators:
\[ 4 \cdot 3 + k^2 = 8k \]
This simplifies to:
\[ k^2 - 8k + 12 = 0 \]

6. **Solve the Quadratic Equation**:
We can factor the quadratic:
\[ (k - 2)(k - 6) = 0 \]
Thus, the solutions for \( k \) are:
\[ k = 2 \quad \text{or} \quad k = 6 \]

### Final Answer:
The values of \( k \) are \( 2 \) and \( 6 \).