Question

question 25 · 1 point
what is the standard form of the equation of the parabola with the focus (2, - 1) and the directrix y = 3?
select the correct answer below:
o (y - 3)^2 = 6(x - 1/2)
o (y + 1)^2 = - 2(x - 5/2)
o (x - 2)^2 = - 8(y - 1)
o (x - 2)^2 = 8(y - 1)

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus $(2,-1)$ and the point on the directrix directly above or below the focus. The directrix is $y = 3$. The $x$ - coordinate of the vertex is the same as the $x$ - coordinate of the focus, $x=2$. The $y$ - coordinate of the vertex is $y=\frac{-1 + 3}{2}=1$. So the vertex $(h,k)=(2,1)$.

Step2: Determine the value of $p$

The distance $p$ between the vertex and the focus (or the vertex and the directrix) is $|1-(-1)| = 2$. Since the focus $(2,-1)$ is below the vertex $(2,1)$, the parabola opens downwards and $p=-2$.

Step3: Use the standard form equation

The standard form of a parabola that opens up or down is $(x - h)^2 = 4p(y - k)$. Substituting $h = 2,k = 1,p=-2$ into the equation, we get $(x - 2)^2=4\times(-2)(y - 1)=(x - 2)^2=-8(y - 1)$.

Answer:

C. $(x - 2)^2=-8(y - 1)$