Question

question
what is the standard form of the equation of the parabola with the focus (6, -1) and the directrix x = -10? 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 $(x_f,y_f)=(6, - 1)$ and the point on the directrix directly opposite the focus. The directrix is $x=-10$. The $x$ - coordinate of the vertex $h=\frac{6+( - 10)}{2}=\frac{6 - 10}{2}=-2$, and the $y$ - coordinate of the vertex $k=-1$. So, $(h,k)=(-2,-1)$.

Step2: Find the value of $p$

The distance between the vertex and the focus (or the vertex and the directrix) is $p$. The $x$ - coordinate of the focus is $x_f = 6$ and the $x$ - coordinate of the vertex is $h=-2$. So, $p=6-( - 2)=8$.

Step3: Write the equation of the parabola

Substitute $h=-2$, $k = - 1$, and $p = 8$ into the standard form $(y - k)^2=4p(x - h)$. We get $(y+1)^2=4\times8(x + 2)$, which simplifies to $(y + 1)^2=32(x + 2)$.

Answer:

$(y + 1)^2=32(x + 2)$