Find the equation of a parabola given the following information:
Focus (5, 3) and directrix: x = -5
Since the directrix is a vertical line, we know that the parabola opens horizontally. Let (x, y) be any point on the parabola. The distance between this point and the focus is given by the distance formula:
sqrt((x - 5)^2 + (y - 3)^2)
The distance between this point and the directrix is the horizontal distance between the point and the line x = -5, which is simply |x + 5|.
By definition, these two distances are equal, so we can set them equal to each other and square both sides to eliminate the square root:
(x + 5)^2 = (x - 5)^2 + (y - 3)^2
Expanding the squares and simplifying, we get:
4x = y^2 - 6y + 34
This is the equation of the parabola in standard form.
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