Question

the focus of a parabola is located at (4,0), and the directrix is located at x = -4. which equation represents the parabola? \\(\bigcirc\\) \\(y^2 = -x\\) \\(\bigcirc\\) \\(y^2 = x\\) \\(\bigcirc\\) \\(y^2 = -16x\\) \\(\bigcirc\\) \\(y^2 = 16x\\)

Explanation:

Step1: Recall the standard form of a parabola

For a parabola that opens to the right or left, the standard form is \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex and \(p\) is the distance from the vertex to the focus (and also to the directrix). The focus is \((h + p, k)\) and the directrix is \(x = h - p\).

Step2: Determine the vertex

The vertex is halfway between the focus \((4,0)\) and the directrix \(x=-4\). The x - coordinate of the vertex \(h=\frac{4+( - 4)}{2}=0\), and the y - coordinate \(k = 0\) (since the focus has \(y = 0\) and the parabola is symmetric about the x - axis). So the vertex is \((0,0)\).

Step3: Find the value of \(p\)

The focus is \((h + p,k)=(4,0)\) and \(h = 0,k = 0\), so \(h + p=4\), substituting \(h = 0\) we get \(p = 4\). Since the focus is to the right of the vertex (\(p>0\)), the parabola opens to the right.

Step4: Write the equation of the parabola

Using the standard form \((y - k)^2=4p(x - h)\) with \(h = 0,k = 0,p = 4\), we have \((y-0)^2=4\times4\times(x - 0)\), which simplifies to \(y^{2}=16x\).

Answer:

\(y^{2}=16x\) (corresponding to the option: \(y^{2}=16x\))