Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = −1?
(Please explain too)

f(x) = −one fourth x^2

f(x) = one fourth x^2

f(x) = −4x^2

f(x) = 4x^2

Question

Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = −1?
(Please explain too)

f(x) = −one fourth x^2

f(x) = one fourth x^2

f(x) = −4x^2

f(x) = 4x^2

hero · Answer

To find the equation of the parabola plug points (0,1) and (x, -1) into the formula

$(x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2$

to get:

$(x - 0)^2 + (y - 1)^2 = (x - x)^2 + (y - (-1))^2$

Then simplify:

$x^2 + (y - 1)^2 = 0 + (y + 1)^2$

And expand:

$x^2 + y^2 - 2y + 1 = y^2 + 2y + 1$

From here, continue isolating y.