Question

If $\frac{a-x}{px}=\frac{a-y}{qy}=\frac{a-z}{rz}$ and p,q,r be in A.P. then prove that x,y,z are in H.P

Answer 1

2q = p+ r $\because$ p,q,r are in A.P. ...(1)
Let $\frac{a-x}{px}=\frac{a-y}{py}=\frac{a-z}{pz}=ksay....\left(2\right)$
$p=\frac{a-x}{kx}=\frac{1}{k}(\frac{a}{x}-1)etc.$

Since p,q,r are in A.P. $\therefore \frac{a}{x},\frac{a}{y},\frac{a}{z}$ are also in
A.P. or x,y,z are in H.P.