Identify the relation between the coordinates!

The line joining origin to (a,b) and origin to (c,d) are perpendicular, so product of their slopes= $-1$ .

$\frac {b}{a}×\frac {d}{c}=-1$ implies $\boxed {bd=-ac}$

ac+bd = 0 . dot product of vectors is zero .

(a, b) , (c, d), (0, 0) form a triangle. Let A (a, b) , B(c, d) and O (0, 0).

Area of triangle ABO=1/2 |a(d-0) + c (0-b) + 0 (b-d)| =1/2 (ad-bc) -(1)

Since A(a, b) & B(c, d) subtends a rt.angle at the origin O (0, 0) , therefore ABO is a rt.triangle .

AO=(a^2 + b^2)^1/2 [using distance formula]

BO=(c^2 +d^2)^1/2

Area of ABO= 1/2 × base x height = 1/2 ( AO x BO) -(2)

From (1) & (2), (ad - bc)^2 = (a^2 + b^2) (c^2 + d^2)

or, -2abcd = a^2c^2 + b^2d^2

or, (ac + bd)^2 =0

or, ac + bd=0

ac= -bd is the relation between (a, b) & (c, d).