Question

If $a,b,c$ are in A.P. $p,q,r$ are in H.P. and $ap,bq,cr$ are in G.P., then :

• A. $\frac{p}{r}+\frac{r}{p}=\frac{c}{a}+\frac{a}{c}$
• B. $\frac{p}{q}+\frac{q}{p}=\frac{c}{a}+\frac{a}{c}$
• C. $\frac{p}{r}+\frac{r}{p}=\frac{b}{c}+\frac{c}{b}$
• D. none of these

Answer 1

The correct option is A $\frac{p}{r}+\frac{r}{p}=\frac{c}{a}+\frac{a}{c}$

a,b,c are in A.P

$\therefore 2b=a+c$ ______ (1) $\therefore b=\frac{a+c}{2}$

p,q,r in H.P

$\therefore \frac{1}{p}+\frac{1}{r}=\frac{2}{q}$ ______ (2) $\therefore q=\frac{2pr}{r+p}$

ap, bq, cr in C.P

$(bq{)}^2=(ap\left)\right(rc)$ ________ (3)

putting b and q from (1) and (2) in (3)

$\left[\right(\frac{a+c}{2}\left)\right(\frac{2pr}{r+p}){]}^2=aprc$

$\frac{(a+c{)}^2}{(r+p{)}^2}\times p^2{r}^2=aprc$

$\therefore \frac{pr}{(r+p{)}^2}=\frac{ac}{(a+c{)}^2}$

$\therefore \frac{a^2+2ac+c^2}{ac}=\frac{r^2+2rp+p^2}{pr}$

$\therefore \frac{a^2}{ac}+\frac{2ac}{ac}+\frac{c^2}{ac}=\frac{r^2}{pr}+\frac{2rp}{rp}+\frac{p^2}{rp}$

$\frac{a}{c}+\frac{c}{a}=\frac{r}{p}+\frac{p}{r}$ LHS=RHS