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Question

If the point P(x,y) is equidistant from the points A(a+b,ba) and B(ab,a+b). Prove that bx=ay.

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Solution

Since the point P(x,y) is equidistant from the points A(a+b,ba)(x1,y1) and B(ab,a+b)(x2,y2).

Using distance formula,

Distance Formula =(xx1)2+(yy1)2

PA=(x(a+b))2+(y+(ba))2

PB=(x(ab))2+(y+(a+b))2

Therefore, PA=PB

(x(a+b))2+(y+(ba))2=(x(ab))2+(y+(a+b))2

Squaring on both the sides,

(x(a+b))2+(y+(ba))2=(x(ab))2+(y+(a+b))2

x2+(a+b)22x(a+b)+y2+(ba)22y(ba)=x2+(ab)22x(ab)+y2+(a+b)22y(b+a)

2x(a+b)2y(ba)=2x(ab)2y(b+a)

2ax2bx2by+2ay=2ax+2bx2by2ay

4bx=4ay

bx=ay

Hence proved.


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