Find an equation for the parabola with focus (9, 3) and directrix y = 9. A) x² - 18x + 12y + 9 = 0 B) x² - 18x - 12y + 9 = 0 C) y² - 18y + 12x + 9 = 0 D) y² - 18y - 12x + 9 = 0

Question

Quizplus · Accepted Answer

The equation of a parabola can be derived using the definition that each point on the parabola is equidistant from the focus and the directrix. Given the focus (9, 3) and the directrix y = 9, the vertex is halfway between the focus and the directrix, which places it at (9, 6). The distance from the vertex to the focus (and also to the directrix) is 3, indicating a vertical parabola that opens downwards because the focus is below the directrix. The standard form of a parabola opening vertically is $(x-h)^2 = 4p(y-k)$, where (h, k) is the vertex and p is the distance to the focus or directrix. Substituting the given values, we get $(x-9)^2 = -12(y-6)$, which simplifies to $x^2 - 18x + 12y + 9 = 0$, matching choice A.

Find an Equation for the Parabola with Focus (9, 3)