Question 203940
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Find the equation of the line with points equidistant from (5,-2) and (4,3).
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        As an alternative to the geometric solution by @Theo,  below is simple algebraic solution.



<pre>
Let (x,y) be the point equidistant from these given points (5,-2) and (4,3).


It means that the distances from (x,y) to these points are equal.

Hence, the squares of distances are equal.


So, we write

    {{{(x-5)^2}}} + {{{(y-(-2))^2}}} = {{{(x-4)^2}}} + {{{(y-3)^2}}}.

or

    {{{(x-5)^2}}} + {{{(y+2)^2}}} = {{{(x-4)^2}}} + {{{(y-3)^2}}}.


Now we simplify it

    x^2 - 10x + 25 + y^2 + 4y + 4 = x^2 - 8x + 16 + y^2 - 6y + 9,

    -10x + 4y + 29 = -8x - 6y + 25,

    (-10x + 8x) + (4y + 6y) = 25 - 29,

    -2x + 10y = -4,

     -x +  5y = -2,

      x -  5y = 2.
</pre>

Solved.