Problem 28 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((2,4) ;\) Directrix: \(x=-4\)

Step-by-Step Solution

Answer

The standard form of the equation for the parabola is \( (y-4)^2 = 12(x+1) \)

Step 1: Determine the vertex of the parabola

The vertex of the parabola is the midpoint of the focus and a point on the directrix directly across from the focus. Since the focus is (2,4) and the directrix is x=-4, we can take any point on the directrix (like (-4,4)) and find the midpoint. The vertex (h,k) will then be \(([2 +(-4)]/2 , [4+4]/2) = (-1,4)\)

Step 2: Find the value of p

The value of p can be found by the formula \(p = h - x\), where x is any x-value on the directrix. For the directrix \( x = -4\), and for h from vertex coordinates, \(h = -1\), so, \(p = -1 - (-4) = 3\)

Step 3: Construct the parabolic equation

Now we can plug in the values of h, k and p into the standard formula \((y-k)^2 = 4p(x-h)\), to get \((y-4)^2 = 4*3 (x-(-1))\), simplifying this results in \((y-4)^2 = 12(x+1)\)