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. ax = by
• B. ax² = by
• C. ay = bx
• D. ay² = bx

Answer 1

The correct option is C

ay = bx

Let points A(a + b, a - b) and B(a - b, a + b) be equidistant from point P (x, y ).
By, distance formula PA = PB,
$\sqrt{(x-(a+b){)}^2+(y-(b-a){)}^2}=\sqrt{(x-(a-b){)}^2+(y-(a+b){)}^2}$
Squaring both sides,
$(x-(a+b){)}^2+(y-(b-a){)}^2=(x-(a-b){)}^2+(y-(a+b){)}^2\Rightarrow (x-(a+b){)}^2-(x-(a-b){)}^2=(y-(a+b){)}^2-(y-(b-a){)}^2$
We can use this identity $a^2-b^2=(a+b)(a-b)$ to solve the above equation. After solving, we will get:
ay = bx