Question 992821
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This point has coordinates  (x,x)  with the unknown  x,

and the equation to find  x  is the equality of distances 


from  (x,x)  to  (8,9):   {{{sqrt((8-x)^2 + (9-x)^2))}}},     and


from  (x,x) to  (1, -10):   {{{sqrt((1-x)^2 + (-10-x)^2))}}}.


This equation is  (after squaring the square roots):


{{{(8-x)^2 + (9-x)^2)}}} = {{{(1-x)^2 + (-10-x)^2)}}}, or


{{{64 -16x + x^2}}} + {{{81 - 18x + x^2}}} = {{{1 - 2x + x^2}}} + {{{100 + 20x + x^2}}}.


Simplify and solve it.


64 + 81 -1 - 100 = 16x + 18x - 2x + 20x,


44 = 52x


x = {{{44/52}}} = {{{11/13}}}.


<U>Answer</U>. &nbsp;The point is &nbsp;({{{11/13}}}, {{{11/13}}}).