SOLUTION: find the equation describing all points equidistant from the x-axis and the point (0,2)

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Question 1137032: find the equation describing all points equidistant from the x-axis and the point (0,2)
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!
find an equation describing all points equidistant from the x-axis and the point (, ).
First, see if you can sketch a picture of what this curve ought to look like.
For a point (, ) that is on the curve, explain why .
note:
is the distance from (, ) to the x-axis
is the distance from (, ) to the point (, )
distances must be equal because is given:all points equidistant from the x-axis and the point (,)

Square both sides of this equation and solve for. Identify the curve.



......complete square for





=>hyperbola



Answer by ikleyn(52795)   (Show Source): You can put this solution on YOUR website!
.

            The solution by @MathLover1 is incorrect.

            It is incorrect,  because the formula for the distance from the point (0,2) to the point  (x,y) is   ,  and not   .

            The final curve should be a parabola,  not a hyperbola.

            I know it very well,  since I solved similar problems several times at this forum.

            The correct solution is below.


Solution 

The distance from  (, ) to the x-axis is  |y|.

 
The distance from   (, ) to the point (, ) is  .


The base equation is


|y| = .


Square both sides of this equation and solve for.  

 = 

 = 

 = 

4y =  

y =     <==========>  parabola with the branches upward; the vertex at the point (0,1)



    


                Plot  y =  + 1


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