Question

question
what is the standard form of the equation of the parabola with the focus (-2, -7/2) and the directrix y = -5/2?
select the correct answer below:
(x + 2)^2 = -2(y + 3)
(x + 2)^2 = -4(y + 3)
(x + 2)^2 = 2(y + 3)
(x + 2)^2 = 4(y + 3)

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus and the point on the directrix directly below (or above) the focus. The x - coordinate of the vertex is the same as the x - coordinate of the focus, i.e., $x=-2$. The y - coordinate of the vertex $k$ is $\frac{-\frac{7}{2}+(-\frac{5}{2})}{2}=\frac{-\frac{7 + 5}{2}}{2}=\frac{-6}{2}=-3$. So the vertex is $(-2,-3)$.

Step2: Determine the value of $p$

The distance between the vertex $(h,k)=(-2,-3)$ and the focus $(-2,-\frac{7}{2})$ is $p$. Using the formula for the distance in the y - direction, $p=-\frac{7}{2}-(-3)=-\frac{7}{2}+3=-\frac{7}{2}+\frac{6}{2}=-\frac{1}{2}$.

Step3: Write the standard form of the parabola equation

The standard form of a parabola with a vertical axis of symmetry is $(x - h)^2 = 4p(y - k)$. Substituting $h=-2$, $k = - 3$, and $p=-\frac{1}{2}$ into the equation, we get $(x+2)^2=4\times(-\frac{1}{2})(y + 3)$. Simplifying gives $(x + 2)^2=-2(y + 3)$.

Answer:

$(x + 2)^2=-2(y + 3)$