If x1, x2, x3  and y1, y2, y3  are both in GP with the same common ratio, then the points (x1, y1), (x2, y2)  and (x3, y3)

If x1, x2, x3  and y1, y2, y3  are in GP.Then let x2=rx1, x3=r2x1And y2=ry1, y3=r2y1With common ratio r , then the points are(x1, y1), (rx1, ry1)  and (r2x1, r2y1) .Now x1y11x2y21x3y31=x1y11rx1ry11r2x1r2y11=x1y1111rr1r2r21=x1y1(0)=0(Since two columns are identical)Thus these points lie on a straight line.Alternate SolutionLet x1=a⇒x2=ar  and x3=ar2And y=b⇒y2=br  and y3=br2Let the points are A(a, b), B(ar, br)  and C(ar2, br2) .Now slope of AB=b(r−1)a(r−1)=baAnd slope of BC=b(r2−r)a(r2−r)=ba∵   slope of AB= slope of BC⇒  AB∥BCBut B  is a common point.∴  A, B  and C  are collinear.i.e., the points (x1, y1), (x2, y2)  and (x3, y3)  lie on a straight line.