Derive the equation of the parabola with a focus at (2, −1) and a directrix of y = −one half.

Asked by maham237 @ 03/12/2020 in Mathematics viewed by 333 People

f(x) = −(x + 2)2 − three fourths
f(x) = (x + 2)2 + three fourths
f(x) = −(x − 2)2 + three fourths
f(x) = −(x − 2)2 − three fourths

Answered by maham237 @ 03/12/2020

Basically, the parabola has to have all points that are equidistant from the focus and the directrix, the directrix being a horizontal line, and the focus being a point given. To derive an equation from this you need to use the distance formula which I'm guessing you already know because you're already in precalc.
The gist of it is that we have a random point on the parabola (x,y), and the point (x,y) will be equidistant from both the focus and the directrix. If we use the distance formula, you get something like this:
$\sqrt{(y-(- \frac{1}{2} ))^2} = \sqrt{(x-2)^2+(y-(-1)^2}$
The square root of y-(-1/2) coming from the directrix, and the righthand side of the equal sign being derived from the focus.
All you need to do is simplify now!
$(y+\frac{1}{2})^2 = (x-2)^2+(y+1)^2 \\\ y^2+y+ \frac{1}{4} = x^2-4x+4 + y^2+2y+1 \\\ -y-\frac{3}{4} = x^2-4x+4 \\\ -y-\frac{3}{4} = (x-2)^2 \\\ -y = (x-2)^2+\frac{3}{4} \\\ y = -(x-2)^2-\frac{3}{4}$

Hope I helped!

Derive the equation of the parabola with a focus at (2, −1) and a directrix of y = −one half.

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