Question

1. find the equation of the parabola with the focus, f (0, 3) and a directrix of y = -5.

Explanation:

Step1: Find the vertex

The vertex \((h,k)\) of a parabola is the mid - point between the focus \((0,3)\) and the directrix \(y = - 5\).
Using the mid - point formula for the \(y\) - coordinate (since the \(x\) - coordinate of the focus is \(0\) and the directrix is a horizontal line \(y=-5\), the \(x\) - coordinate of the vertex \(h = 0\)).
\(k=\frac{3+( - 5)}{2}=\frac{3 - 5}{2}=-1\). So the vertex is \((0,-1)\).

Step2: Find the value of \(p\)

The distance \(p\) between the vertex \((0,-1)\) and the focus \((0,3)\) (or the vertex and the directrix). Using the formula \(p=y_{focus}-y_{vertex}\) (since the parabola opens upwards as the focus is above the vertex).
\(p = 3-( - 1)=4\).

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 \(p = 4\) into the equation.
\((x - 0)^{2}=4\times4(y+1)\)

Answer:

\(x^{2}=16(y + 1)\)