Alg6.7

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Find the relation between x and y,if the point (x,y) is equidistant from (7,-6),(-3,4)\,

The distance between (x,y) and the first point is

{\sqrt  {(x-7)^{2}+(y+6)^{2}}}\,

{\sqrt  {x^{2}+49-14x+y^{2}+36+12y}}\,

{\sqrt  {x^{2}+y^{2}-14x+12y+85}}\,

Let this expression be 1.

The distance between (x,y) and the second point is

{\sqrt  {(x+3)^{2}+(y-4)^{2}}}\,

{\sqrt  {x^{2}+9+6x+y^{2}+16-8y}}\,

{\sqrt  {x^{2}+y^{2}+6x-8y+25}}\,

Let this expression be 2.

As per the given condition,equating both the expressions,

{\sqrt  {x^{2}+y^{2}-14x-12y+85}}={\sqrt  {x^{2}+y^{2}+6x-8y+25}}\,

Squaring on both sides,we get

x^{2}+y^{2}-14x+12y+22=x^{2}+y^{2}+6x-8y+25\,

cancelling the terms which are common on both sides,

-14x-6x+12y+8y=-60\,

-20x+20y=-60\,

x-y=3\,

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