Question 550844
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The points A(-1,2), B(x,y) and C(4,5) are such that BA=BC. Find a linear relation between x and y.
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        This problem has a nice Algebra solution.



<pre>
The square of the length of AB is 

    |AB|^2 = (x-(-1))^2 + (y-2)^2 = (x+1)^2 + (y-2)^2 = x^2 + 2x + 1 + y^2 - 4y + 4 = x^2 + 2x + y^2 - 4y + 5.



The square of the length of BC is 

    |BC|^2 = (x-4)^2 + (y-5)^2 = x^2 - 8x + 16 + y^2 - 10y + 25 = x^2 - 8x + y^2 - 10y + 41.



The condition  |BA| = |BC|  is the same as  |AB|^2 = |BC|^2.  It gives this equation 

    x^2 + 2x + y^2 - 4y + 5 = x^2 - 8x + y^2 - 10y + 41.



Combine like terms.  The final equation is

    10x + 6y = 36,

or

    5x + 3y = 18,

or

    y = {{{(-5/3)*x + 6}}}.    <<<---===  <U>ANSWER</U>
</pre>

Solved.