Question

One diagonal of a square is the intercept of the line x/a + y/b = 1 between the axes. Find the coordinates of other two vertices and hence, prove that if two opposite vertices of a square move on two perpendicular lines, the other two vertices also move on perpendicular lines.

Answer 1

See Fig.

The coordinates of centre E are (a/2,b/2). The slope of AC  = -b/a

The slope of DB = a/b ⇒ a/b sinθ = a/(√a² + b²)

⇒cosθ = b/(√a² + b²)

Length AE = EC = EB = ED = 1/2 AC = 1/2√a² + b²

Using the distance form for DB, we get

Thus, the coordinates of other two vertices are thus

Now, given that A and C move on perpendicular lines (axes). For the coordinates (a + b/2,b + a/2),we have the locus x = y and for (a - b/2,b - a/2) the locus is x = −y. These are also perpendicular lines.