Question

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

Answer 1

Answer:

Given:

(x,y) is equidistant from the points A(5,1) and B(1,5).

To prove: x=y

Proof;

Let the arbitrary point be O(x,y).

Thus,

According to question,

=> AO=OB

[ note: as per distance formula,

the distance between two points

(x1,y1) and (x2,y2) is given by;

d= √{(x2-x1)^2 + (y2-y1)^2}

Thus , here we have;

AO= √{(x-5)^2 +(y-1)^2} and

OB= √{(x-1)^2 +(y-5)^2}. ]

=> √{(x-5)^2 +(y-1)^2}=√{(x-1)^2 +(y-5)^2}

now squaring, both sides,

we have,

=> (x-5)^2 + (y-1)^2 = (x-1)^2 + (y-5)^2

=> x^2 + 25 - 10x + y^2 +1 - 2y

= x^2 + 1 - 2x + y^2 + 25 - 10y

=> -10x - 2y = -2x - 10y

=> 10x - 2x = 10y - 2y

=> 8x = 8y

=> x = y

Hence proved.