Problem 30 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((7,-1) ;\) Directrix: \(y=-9\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \(x = 4(y + 5)^2 + 7\)

Step 1: Calculate the vertex

The vertex of the parabola is midway between the focus and the directrix. Since the directrix is \(y = -9\) and the y-coordinate of the focus is -1, the y-coordinate of the vertex is the average of these two values. So, \(y = (-9 + (-1))/2 = -5\). The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 7. So the vertex is \( (7, -5) \)

Step 2: Calculate the value of p

The value of p is the distance from the vertex to the focus. In this case, since the vertex and the focus have the same x-coordinate, p is simply the difference in their y-coordinates, which is \( -1 - (-5) = 4 \)

Step 3: Write the equation of the parabola

The general form of the equation of a parabola with a horizontal axis is \(x = py^2 + h\). We found earlier that p = 4 and h = 7, so the standard form of the equation of the parabola is \(x = 4(y + 5)^2 + 7 \)