Question

given a parabola with focus (0,2) and directrix y = - 2, which of the following represents the correct equation of this parabola?

o y=\frac{1}{2}x^{2}

o y = -\frac{1}{2}x^{2}

o y=\frac{1}{8}x^{2}

o y=\frac{1}{4}x^{2}

Explanation:

Step1: Recall the formula for a parabola

The standard - form equation of a parabola with its vertex at the origin \((0,0)\) and opening upwards or downwards is \((x - h)^2=4p(y - k)\), where \((h,k)\) is the vertex of the parabola and \(p\) is the distance between the vertex and the focus (or the vertex and the directrix). The vertex of a parabola is the mid - point between the focus and the directrix.

Step2: Find the vertex

The focus is \((0,2)\) and the directrix is \(y=-2\). The \(x\) - coordinate of the vertex is \(x = 0\) (since the focus and directrix are symmetric about the \(y\) - axis). The \(y\) - coordinate of the vertex \(k=\frac{2+( - 2)}{2}=0\). So the vertex \((h,k)=(0,0)\).

Step3: Calculate the value of \(p\)

The distance \(p\) between the vertex \((0,0)\) and the focus \((0,2)\) is \(p = 2\) (or the distance between the vertex \((0,0)\) and the directrix \(y=-2\)).

Step4: Write the equation of the parabola

Substitute \(h = 0\), \(k = 0\), and \(p = 2\) into the equation \((x - h)^2=4p(y - k)\). We get \(x^{2}=4\times2y\), which simplifies to \(x^{2}=8y\), or \(y=\frac{1}{8}x^{2}\).

Answer:

\(y=\frac{1}{8}x^{2}\) (corresponding to the third option in the multiple - choice list)