Final answer:
To find the equation of a parabola with a focus at (-5, 0) and directrix x = 5, equate the distances from a point (x, y) to the focus and directrix. By solving, we obtain the parabola's equation: y^2 = -20x.
Step-by-step explanation:
To write the equation of a parabola given a focus at (-5, 0) and a directrix of x = 5, we should recall that the parabola is the set of points equidistant from the focus and the directrix. For any point (x, y) on the parabola, the distance to the focus (-5, 0) must equal the distance to the directrix x = 5.
Using the distance formula for the distance between (x, y) and the focus (-5, 0), we have:
√((x + 5)^2 + (y - 0)^2)
The distance between (x, y) and the directrix x = 5 is simply the absolute value of the difference between the x-coordinates, which is |x - 5|.
Setting these distances equal to each other, we have:
√((x + 5)^2 + y^2) = |x - 5|
Since the distances are positive, we can square both sides to get rid of the square root and the absolute value, resulting in:
(x + 5)^2 + y^2 = (x - 5)^2
Expanding and simplifying, we get the equation of the parabola:
y^2 = -20x