If the point P(X,y) is equidistant from the points A(a+b,b-a) and B(a-b,a+b),prove that bx=ay.

Given that:

Distance between the points P(x, y) and A(a+b, b-a) & B(a-b, a+b) are equal

To prove: bx = ay

Proof: From the given data

PA=PB
On squaring both sides we get,

PA2=PB2

Using the distance formula the equation can be written as
={x-(a+b)}2+{y-(b-a)}2={x-(a-b)}2+{y-(a+b)}2

= x2+(a+b)2-2x(a+b)+y2+(b-a)2-2y(b-a)y=x2+(a-b)2-2x(a-b)+y2+(a+b)2-2y(a+b)

= 2x(a-b)-2x(a+b)=2y(b-a)-2y(a+b)

= 2x{a-b-a-b}=2y{b-a-a-b}

= 2x(-2b)=2y(-2a)

= bx=ay

Hence Proved

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