Question

A, B , C are the points (a, p), (b,q) and (c,r) respectively such that a, b, c are in A. P. and p, q ,r in G.P. If the points are collinear then prove that p = q = r

Answer 1

Since the points are collinear we must have $\Delta =0$
$\therefore \left|\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right|=0$

Since a, b, c are in A.P . Apply $R_1+R_3-2R_2$
$\therefore \left|\begin{array}{ccc}0& p+r-2q& 0\\ b& q& 1\\ c& r& 1\end{array}\right|=0$

$\therefore (p+r-2q)(b-c)=0$ $\therefore 2q=p+r$

but $q^2=pror{\left(\frac{P+r}{2}\right)}^2=pr$

or $(p-r{)}^2=0$ $\therefore p=r$

$\therefore 2q=2porp=q\therefore p=q=r$