Problem 20 - College Algebra

Question

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((-10,0) ;\) Directrix: \(x=10\)

Step-by-Step Solution

Answer

The standard form of the equation of the parabola is \( y^2 = -40x \).

Step 1: Determine the direction of parabola

From given condition, it is clear that the parabola opens towards the left because the focus \( (-10,0) \) is to the left of the directrix \( x=10 \).

Step 2: Find the vertex

The vertex of a parabola is located halfway between the focus and the directrix. Using this information, the vertex can be calculated as: \( ((-10+10)/2, 0) = (0,0) \).

Step 3: Identifying elements to write down the formula

For a parabola that opens left or right, the standard form of the equation is \( (y - k)^2 = 4p (x - h). \) Here, (h,k) are the coordinates of the vertex and p is the distance from the vertex to the focus or from the vertex to the directrix.

Step 4: Substituting the values

By substituting the vertex \((h, k) = (0, 0)\) and the distance \(p = -10\) (negative because parabola opens to the left), the standard form of the equation of parabola becomes: \( (y - 0)^2 = 4*(-10)*(x - 0). \)

Step 5: Simplify the equation

Finally, simplifying the equation yields: \( y^2 = -40x \).