Question

derive the equation of a parabola with a focus at (4, 6) and a directrix at y = -4. enter your answers in the boxes. be sure to include the correct signs. y = □(x □)^2 □

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus $(4,6)$ and the point on the directrix directly below/above the focus. The $x$ - coordinate of the vertex is the same as the $x$ - coordinate of the focus, $x = 4$. The $y$ - coordinate of the vertex is $\frac{6+( - 4)}{2}=\frac{6 - 4}{2}=1$. So the vertex $(h,k)=(4,1)$.

Step2: Find the value of $p$

The distance $p$ between the vertex and the focus (or the vertex and the directrix) is $p=6 - 1 = 5$ (since the focus is above the vertex, $p>0$).

Step3: Use the standard form of the parabola equation

The standard form of a parabola that opens up or down is $y=a(x - h)^2+k$, where $a=\frac{1}{4p}$. Since $p = 5$, then $a=\frac{1}{20}$.
The equation of the parabola is $y=\frac{1}{20}(x - 4)^2+1$.

Answer:

$y=\frac{1}{20}(x - 4)^2+1$