Question

If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), then which of the following is correct

• A. by = ax
• B. bx = ay
• C. bx = a
• D. x = ay

Answer 1

The correct option is B

bx = ay

Let, P(x,y), Q(a + b, b - a) and R(a - b, a + b) are the given points.

Then, according to question:

PQ = PR (Given)

$\Rightarrow \sqrt{{\{x-(a+b\left)\right\}}^2+{\{y-(b-a\left)\right\}}^2}=\sqrt{{\{x-(a-b\left)\right\}}^2+{\{y-(a+b\left)\right\}}^2}$

Squaring both sides,

$\Rightarrow {\{x-(a+b\left)\right\}}^2+{\{y-(b-a\left)\right\}}^2={\{x-(a-b\left)\right\}}^2+{\{y-(a+b\left)\right\}}^2$

$\Rightarrow x^2-2x(a+b)+{(a+b)}^2+y^2-2y(b-a)+{(b-a)}^2$

$=x^2+{(a-b)}^2-2x(a-b)+y^2-2y(a+b)+{(a+b)}^2$

$\Rightarrow -2x(a+b)-2y(b-a)=-2x(a-b)-2y(a+b)$

$\Rightarrow ax+bx+by-ay=ax-bx+ay+by$

$\Rightarrow 2bx=2ay\Rightarrow bx=ay$