Let P be a point on the perpendicular bisector of the segment joining (2, 3) and (6, 5). If the abscissa and the ordinate of P are equal, find the coordinates of P.

Answers (1)

Let the point =       $$ P(x, y)$$ is equal to coordinate then we have $$ y= x $$

therefore, the coordinate of P =$$ (x, x )

Now, let A and B denote the point (2,3) and (6, 5) Since P is equidistant from A and B

We get,   $$ PA^2 = PB^2

 i.e. 

                        $$ (x-2)^2 + (x-3)^2 - (x - 6)^2 + (x -5)^2 $$\$$ $$ x^2 -4x + 4 + x^2 - 6x + 9 = x^2 - 12 x + 36 + x^2 - 10 x + 25$$\$$ $$ 2x^2 -10x + 13 = 2x^2 - 22x +61$$\$$ $$ 22x - 10x = 61 -13$$\$$ $$ 12x = 48$$\$$ $$ x = frac4812 = 4 $$\$$

Hence, the coordinate of P = $$ (4,4)

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