Question 1137032
.


            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    {{{sqrt(x^2 + (y-2)^2)}}},   and not    {{{sqrt(x^2 - (y-2)^2)}}}.


            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.



<U>Solution</U>


<pre>
The distance from  ({{{x}}}, {{{y}}}) to the x-axis is  |y|.

 
The distance from   ({{{x}}}, {{{y}}}) to the point ({{{0}}}, {{{2}}}) is  {{{sqrt(x^2 + (y - 2)^2)}}}.


The base equation is


|y| = {{{sqrt(x^2 + (y - 2)^2)}}}.


Square both sides of this equation and solve for{{{ y}}}.  

{{{y^2}}} = {{{x^2 + (y - 2)^2}}}

{{{y^2}}} = {{{x^2 + (y^2 - 4y+4)}}}

{{{0}}} = {{{x^2 - 4y + 4}}}

4y =  {{{x^2 + 4}}}

y = {{{(1/4)*x^2 + 1}}}    <==========>  parabola with the branches upward; the vertex at the point (0,1)



    {{{graph ( 400, 400, -10, 10, -5, 10,
             x^2/4 + 1
)}}}


                Plot  y = {{{x^2/4}}} + 1
</pre>