Question

which is the standard form of the equation of the parabola that has a vertex of (3,1) and a directrix of x = -2? (x - 3)^2 = 20(y - 1) (y - 1)^2 = -20(x - 3) (y - 1)^2 = 20(x - 3) (x - 3)^2 = -20(y - 1)

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola with vertex \((h,k)\) and axis of symmetry parallel to the \(x -\)axis is \((y - k)^{2}=4p(x - h)\), where \(p\) is the distance from the vertex to the focus (and also from the vertex to the directrix).

Step2: Determine the values of \(h\), \(k\) and \(p\)

Given vertex \((h,k)=(3,1)\). The directrix is \(x=-2\). The distance \(p\) from the vertex \((3,1)\) to the directrix \(x =- 2\) is \(p=3-(-2)=5\) (since for a parabola \((y - k)^{2}=4p(x - h)\), the formula for the directrix is \(x=h - p\)). Here \(h = 3\), \(k = 1\) and \(4p=20\).

Step3: Substitute \(h\), \(k\) and \(4p\) into the standard form

Substituting \(h = 3\), \(k = 1\) and \(4p = 20\) into the formula \((y - k)^{2}=4p(x - h)\), we get \((y - 1)^{2}=20(x - 3)\)

Answer:

\((y - 1)^{2}=20(x - 3)\) (the third option)