Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the parabola with the given characteristics. Focus: (2,2)\(;\) directrix: \(x=-2\)

Step-by-Step Solution

Answer

The standard form equation of the parabola is \((y-2)^2 = 16(x-2)\)

Step 1: Determine the vertex

The vertex of the parabola is the mid-point of the focus and the directrix. Given the focus is at (2,2) and the directrix at x=-2, the x-coordinate of the vertex would be the average of the x-coordinates of the focus and the directrix, which is \((2-(-2))/2=2\). Since the directrix is the line x = -2 (a vertical line), the y-coordinate of the vertex remains the same as that of the focus. Therefore, the vertex (h,k) is (2,2).

Step 2: Determine the direction

Because the focus of the parabola is to the right of the directrix, the parabola must open to the right. Thus, the standard form of the equation of the parabola must be \((y-k)^2 = 4p(x-h)\).

Step 3: Determine the value of p

The value of p is the distance from the focus to the vertex, which is also the distance from the vertex to the directrix. So, p is the difference of the x-coordinates of the vertex and the directrix, which is \(2 - (-2) = 4\).

Step 4: Plug in the values

Input the found values for h, k, and p into \((y-k)^2 = 4p(x-h)\) to obtain the standard equation, that is \((y-2)^2 = 16(x-2)\).