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Question 722846: write the equation whose graph is the set of all points in the plane equidistant from the given point and the given line F(1,0) and x=-1
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! If you are studying parabolas you might remember that the set of all points in the plane equidistant from a given point and a given line is a curve called a parabola.
You might remember that the given point is called the focus and the given line is called the directrix.
You might remember that the vertex of the parabola is halfway between the focus and the directrix, and you would realize that the vertex would be (0,0) in this case.
You might remember that the equation of the parabola is related to the coordinates of the focus and the vertex.
In that case you may choose to write the equation using a formula you have memorized, or you can read from the book (or class notes).
That might be what is expected.
Otherwise, you could just discover the formula on your own, just like an ancient Greek mathematician would, with a little help from Pythagoras.
CALCULATING LIKE A GREEK:
The distance form a point (x,y) to the line  is 
measured as the length of the horizontal segment from (x,y) to point (-1,y) on line .
The distance from point (x,y) to point F(1,0) is
(Pythagoras says so).
The squares of those distances are equal and are
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USING WHAT YOU KNOW ABOUT PARABOLAS:
The directrix is that vertical line , and the focus is F(1,0),
so the vertex is point V(0,0) halfway between focus and directrix and on the horizontal line  which is the axis of symmetry of the parabola.
The equation of a parabola with vertex (h,k) and a horizontal axis of symmetry is
 and if the focus is (h+c,k), then 
In this case the vertex is (0,0), so  and  ,
and the focus is F(1,0), so  and  ,
so the equation is
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