Question

One diagonal of a square is the portion of the line $\frac{x}{a}+\frac{y}{b}=1.$ which is intercepted between the axes.
Find the co-ordinates of other two vertices of the square. Also prove that if two opposite vertices of a square move on two perpendicular lines, the other two vertices also move on two perpendicular lines.

Answer 1

The slope of the given line is - b/a and the two vertices are clearly on the diagonal BD of the square ABCD. If m be the slope of the line inclined at
an angle of ${45}^{\circ }$ to BD, then
$tan(\pm {45}^{\circ })=\frac{m+\frac{b}{a}}{1-m.\frac{b}{a}}=\pm 1$
$\therefore m=\frac{a-b}{a+b}or\frac{(a+b)}{(a-b)}$
$\therefore ABisy-0=\frac{a-b}{a+b}(x-a)$
$ADisy-b=\frac{a+b}{a-b}(x-0)$
Solving these by subtracting etc. the point A is $(\frac{a-b}{2},\frac{a-b}{2})$
That mid-point of AC and BD is same.
$\therefore Cis(\frac{a+b}{2},\frac{a+b}{2})$