Thank you for the opportunity to help you with your question!

Let ( x_{0} , y_{0} ) be any point on the parabola.

Find the distance between (x_{0} , y_{0}) and the focus. Then find the distance between (x_{0} , y_{0}) and directrix. Equate these two distance equations and the simplified equation in x_{0} and y_{0} is equation of the parabola.

The distance between (x_{0} , y_{0}) and (2, -1) is

sqrt((x-2)^2 + (y-(-1))^2)

(x_{0} , y_{0}) and the directrix, y = -1/2 is

| y_{0}– -1/2|.

Equate the two distance expressions and square on both sides.

(x-2)^2 + (y-(-1))^2) = (y+1/2)^2

answer is D.

Please let me know if you need any clarification. I'm always happy to answer your questions.

Jul 10th, 2015

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