Question

find the equation of the parabola with a focus at (0,7) and a directrix y = - 5. what is the equation? y = (simplify your answer.)

Explanation:

Step1: Find the vertex

The vertex \((h,k)\) of a parabola is the mid - point between the focus \((0,7)\) and the directrix \(y = - 5\).
The formula for the \(y\) - coordinate of the mid - point between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(k=\frac{y_1 + y_2}{2}\). Here \(x_1=x_2 = 0\), \(y_1 = 7\) and \(y_2=-5\).
\(k=\frac{7+( - 5)}{2}=\frac{7 - 5}{2}=1\), and \(h = 0\). So the vertex is \((h,k)=(0,1)\).

Step2: Find the value of \(p\)

The distance \(p\) between the vertex \((0,1)\) and the focus \((0,7)\) (or the vertex and the directrix) is given by \(p=y_{\text{focus}}-y_{\text{vertex}}\) (since the parabola opens upwards).
\(p=7 - 1=6\)

Step3: Use the standard form of the parabola equation

The standard form of a parabola that opens up or down is \((x - h)^2=4p(y - k)\). Substituting \(h = 0\), \(k = 1\) and \(p = 6\) into the equation:
\((x-0)^2=4\times6\times(y - 1)\)
\(x^{2}=24(y - 1)\)
Solve for \(y\):
\(y=\frac{1}{24}x^{2}+1\)

Answer:

\(y=\frac{1}{24}x^{2}+1\)