Question

find the equation for the parabola with the following vertex, focus, and directrix. vertex: (2,4) focus: (3,4) directrix: x = 1 (y - 4)^2 = ?(x - )

Explanation:

Step1: Determine the form of the parabola equation

Since the vertex is \((h,k)=(2,4)\) and the focus \((3,4)\) has the same \(y -\) coordinate as the vertex, the parabola opens to the right. The standard form of a parabola that opens to the right is \((y - k)^{2}=4p(x - h)\), where \((h,k)\) is the vertex and \(p\) is the distance between the vertex and the focus (or the vertex and the directrix).

Step2: Calculate the value of \(p\)

The distance \(p\) between the vertex \((2,4)\) and the focus \((3,4)\) is \(p=\vert3 - 2\vert=1\). Also, the distance between the vertex \((2,4)\) and the directrix \(x = 1\) is \(p=\vert2-1\vert = 1\).

Step3: Substitute \(h\), \(k\), and \(p\) into the standard form

We have \(h = 2\), \(k = 4\), and \(4p=4\times1 = 4\).

Answer:

\((y - 4)^{2}=4(x - 2)\)