Problem 18 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((9,0) ;\) Directrix: \(x=-9\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \( y^2 = 36(x-9) \).

Step 1: Identify the Characteristics of the Parabola

First, identify the properties given. The focus is at (9,0) and the directrix is a vertical line at x=-9.

Step 2: Determine the Vertex and Axis of Symmetry

The vertex will be midway between the focus and directrix, and the axis of symmetry will pass through it. The axis of symmetry can be found by averaging the x-coordinates of the focus and directrix, \( \frac{9 - (-9)}{2} = 9 \). This gives an axis of symmetry and vertex at \( x = 9 \).

Step 3: Find p-value

The value of p represents the distance from the vertex to the focus or from the vertex to the directrix. In this case, the distance from the vertex (9,0) to the focus (0,0) is 9. Therefore, \( p = 9 \).

Step 4: Formulate the Equation

Because the parabola opens to the right (positive x direction), the standard form of the equation will be given as \( (y-k)^2 = 4p(x-h) \), where (h,k) are the coordinates of the vertex. Substituting the values of h, k and p into the equation gives \( (y-0)^2 = 4*9(x-9) \). Simplifying this equation will give the final standard form of the parabola equation.

Step 5: Simplify the Equation

Simplify the equation to obtain the final answer. This gives \( y^2 = 36(x-9) \).