Problem 29 - 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 of the parabola is \(y = 4\left(\frac{1}{12}(x+1)\right)^2 = 3\left(x+1\right)^2\)

Step 1: Determine the vertex of the parabola

Since the parabola is horizontally oriented, with the vertex at the midpoint between the focus and directrix, we can find the coordinates of the vertex \(V(h,k)\) by averaging the x-coordinates of the focus \(F(2,4)\) and directrix \(x=-4\). In other words, \(h = \frac{(2 + (-4))}{2} = -1\). The y-coordinate of the vertex is same as that of the focus, i.e. \(k = 4\). Therefore, the vertex is at \(V(-1,4)\).

Step 2: Determine the distance between vertex and focus or directrix

For a horizontally oriented parabola, the distance from the vertex to the focus (or to the directrix) is given by \(a=\frac{1}{4p}\), where \(p\) is the distance. Therefore, \(p = \|h - x\| \), where \(x\) corresponds to the x-coordinate of the focus or directrix. Using the focus \(F(2,4)\), we get \(p = -(-1 - 2) = 3\).

Step 3: Substitute the vertex and distance into the standard form of a parabola

Substitute the vertex (h, k) and the value of \(p\) (from step 2) into the standard form equation of a parabola \(y=k(a(x-h))^2\) (for a horizontally oriented parabola). Let's remember that \(a = \frac{1}{4p}\), so substituting for \(a\) and \(p\) gives us \(a = \frac{1}{4*3} = \frac{1}{12}\). Now substitute \(h = -1\), \(k = 4\), \(a = \frac{1}{12}\) into \(y=k(a(x-h))^2\), giving \(y = 4\left(\frac{1}{12}(x+1)\right)^2\)