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Question 527312: how do I find the focus of a parabola? please explain in the simplest way possible
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! how do I find the focus of a parabola? please explain in the simplest way possible.
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One standard form of an equation for a parabola showing the focus: (x-h)^2=4p(y-k), with (h,k) being the (x,y) coordinates of the vertex, parabola opens upwards, and the axis of symmetry is a vertical line thru the vertex. The focus or focal point is p units from the vertex on the axis of symmetry. The directrix line is also p units from the vertex on the other side, also, on the axis of symmetry. For example, if the vertex is at the origin, (0,0), The coordinates of the focus would be (0,p) and the directrix will be a line, y=-p
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The three other forms as follows:
(1) (x-h)^2=-4p(y-k) , same as first form except parabola opens downward.
Focus would be (0,-p) and the directrix will be a line, y=p
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(2) (y-k)^2=4p(x-h) , same as first form except parabola opens rightward, axis of symmetry: horizontal line thru vertex
Focus would be (p,0) and the directrix will be a line, x=-p
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(3) (y-k)^2=-4p(x-h) , same as first form except parabola opens leftward, axis of symmetry: horizontal line thru vertex
Focus would be (-p,0) and the directrix will be a line, x=p
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To find the focus you need to find p. If you are given the equation of a parabola, you should put it into one of the four standard forms above and note that the coefficient of the right side of the equation is =4p from which you can find p. P can be determined in other ways such as if the directrix is given or the focal width which is=4p.
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Hope this helps.
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