Question

question what is the standard form of the equation of the parabola with the focus (5, -1) and the directrix x = -9? note that the standard form is (y - k)^2 = 4p(x - h). provide your answer below:

Explanation:

Step1: Find the vertex $(h,k)$

The vertex of a parabola is the mid - point between the focus $(5,-1)$ and the point on the directrix $x = - 9$ that is directly across from the focus. The $y$ - coordinate of the vertex is the same as the $y$ - coordinate of the focus, so $k=-1$. The $x$ - coordinate of the vertex $h$ is $\frac{5+( - 9)}{2}=\frac{5 - 9}{2}=-2$.

Step2: Find the value of $p$

The distance between the vertex and the focus (or the vertex and the directrix) is $p$. The distance between the vertex $(-2,-1)$ and the focus $(5,-1)$ is $p=\vert5-( - 2)\vert = 7$.

Step3: Write the equation of the parabola

Substitute $h=-2$, $k = - 1$, and $p = 7$ into the standard form $(y - k)^2=4p(x - h)$. We get $(y+1)^2=4\times7(x + 2)$, which simplifies to $(y + 1)^2=28(x + 2)$.

Answer:

$(y + 1)^2=28(x + 2)$