Question

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

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola that opens to the right or left is \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex.

Step2: Identify the vertex \((h,k)\)

Given vertex \((2,4)\), so \(h = 2\) and \(k = 4\).

Step3: Calculate the value of \(p\)

The distance between the vertex \((h,k)\) and the focus \((h + p,k)\) (since it opens to the right). Here, \(h + p=3\), \(h = 2\), so \(p=3 - 2=1\).

Step4: Substitute \(h\), \(k\), and \(p\) into the standard form

Substitute \(h = 2\), \(k = 4\), and \(p = 1\) into \((y - k)^2 = 4p(x - h)\). We get \((y - 4)^2=4\times1\times(x - 2)\)

Answer:

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