Question

if the point P (x,y) is equidistant from the points A (5,1) and B (-1,5), prove that x-y=0

Answer 1

may be the answer will be correct

Answer 2

SOLUTION

TO DETERMINE

If the point P (x,y) is equidistant from the point A(5,1) and B(1,5 ) then prove that x - y = 0

FORMULA TO BE IMPLEMENTED

For the given two points $\sf{A( x_1 , y_1) \: \: and \: \: B( x_2 , y_2)}$ the distance between the points

$= \sf{ \sqrt{ {(x_2 -x_1 )}^{2} + {(y_2 -y_1 )}^{2} } }$

EVALUATION

Here the given points are A(5,1) and B(1,5)

Now it is given that the point P (x,y) is equidistant from the point A(5,1) and B(1,5)

Thus we have

$\sf \sqrt{ {(x - 5)}^{2} + {(y - 1)}^{2} } = \sqrt{ {(x - 1)}^{2} + {(y - 5)}^{2} }$

$\sf \implies \: {(x - 5)}^{2} + {(y - 1)}^{2} = {(x - 1)}^{2} + {(y - 5)}^{2}$

$\sf \implies \: {x}^{2} - 10x + 25 + {y}^{2} - 2y + 1 = {x}^{2} - 2x + 1 + {y }^{2} - 10y + 25$

$\sf \implies \: - 10x - 2y + 26 = - 2x - 10y + 26$

$\sf \implies \: - 8x = - 8y$

$\sf \implies \: x = y$

$\sf \implies \: x - y = 0$

Hence proved

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