Find the equation in standard form of the parabola with focus and directrix .

Question

Find the equation in standard form of the parabola with focus  and directrix  .

Quizplus · Accepted Answer

Since the focus and directrix are both vertical, we know that the equation of the parabola will have the form $4p(y-k) = (x-h)^2$, where $(h,k)$ is the vertex and $p$ is the distance between the vertex and focus (and also the distance between the vertex and directrix).

The focus is located at $(0, \frac{5}{2})$, so the vertex is directly below it at $(0, \frac{3}{2})$ (which is $\frac{1}{2}$ units below the directrix at $y=2$). Thus, $k = \frac{3}{2}$.

The directrix is the horizontal line $y=2$, so the distance between the vertex and directrix is $p = \frac{1}{2}$.

Thus, the equation of the parabola is $4 \cdot \frac{1}{2} (y-\frac{3}{2}) = (x-0)^2$, which simplifies to $2y-3 = x^2$. Multiplying both sides by 2 and rearranging, we get $x^2 = 4y-6$, which is the standard form of a vertical parabola. Therefore, the answer is B.

Find the Equation in Standard Form of the Parabola with Focus