If the focus of the parabola $x^2 - ky + 3 = 0$ is $(0, 2)$, then a value of $k$ is/are:
Step-by-Step Solution
Key Concept: Transform the parabola into standard form and use the focus coordinate formula to match the given focus location.
Given parabola $x^2 - ky + 3 = 0$ or $x^2 = k\left(y - \frac{3}{k}\right)$, substitute $X = x$ and $Y = y - \frac{3}{k}$ to get $X^2 = kY$ with focus at $\left(0, \frac{k}{4}\right)$. The original parabola's focus is at $\left(0, \frac{3}{k} + \frac{k}{4}\right) = (0, 2)$. Setting $\frac{3}{k} + \frac{k}{4} = 2$ gives $12 + k^2 = 8k$, so $k^2 - 8k + 12 = 0$, yielding $(k-6)(k-2) = 0$ and thus $k = 2$ or $k = 6$.
Correct Answer: 2,4