Question

Find the distance between $A(a+b,b-a)$ and $B(a-b,a+b)$.

Answer 1

$A(x_1,y_1)=A(a+b,b-a)$ and
$B(x_2,y_2)=B(a-b,a+b)$
$|AB{|}^2={(x_1-x_2)}^2+{(y_1-y_2)}^2$
$={\left[\right(a+b)-(a-b\left)\right]}^2+{\left[\right(b-a)-(a+b\left)\right]}^2$
$={\left(2b\right)}^2-{(-2a)}^2$
$=4b^2-4a^2=4(a^2-b^2)$
$\therefore AB=\sqrt{4(a^2-b^2)}=2\sqrt{(a^2-b^2)}$