Question

A parabola can be drawn given a focus of (−6,−1) and a directrix of y=1. Write the equation of the parabola in any form.

Answer 1

Answer:

y = -(x + 6)²/4

Step-by-step explanation:

Since the equation of a parabola is given as the distance of a point to a given point(focus) being equal to the distance of the point fro a given line (directrix),

and our focus is at (-6,-1) and our directrix is y = 1, then by the distance equation, the distance from each point to the point (x, y) is

√[(x - x)² + (y - 1)²] = √[(x - (-6))² + (y - (-1))²]

√[0² + (y - 1)²] = √[(x + 6)² + (y + 1)²]

√(y² -2y + 1) = √[(x² + 12x + 36) + (y² + 2y + 1)]

squaring both sides, we have

y² - 2y + 1 = x² + 12x + 36 + y² + 2y + 1

collecting like terms, we have

y² - 2y - y² - 2y = x² + 12x + 36 + 1 - 1

simplifying, we have

- 4y = x² + 12x + 36

factorizing the expression in x, we have

-4y = (x + 6)²

y = -(x + 6)²/4