Question

If $\frac { 1 } { p + q } , \frac { 1 } { r + p } , \frac { 1 } { q + r }$ are in A.P., then.

• A. are in A.P.
• B. $p ^ { 2 } , q ^ { 2 } , r ^ { 2 }$ are in A.P.
• C. $\frac { 1 } { p } , \frac { 1 } { q } , \frac { 1 } { r }$ are in A.P.
• D. None of these

Answer 1

Answer: (B)

$p ^ { 2 } , q ^ { 2 } , r ^ { 2 }$ are in A.P.

Answer 2

Since $\frac { 1 } { p + q } , \frac { 1 } { r + q }$ and $\frac { 1 } { q + r }$ are in A.P.

$\therefore$ $\frac { 1 } { r + q } - \frac { 1 } { p + q } = \frac { 1 } { q + r } - \frac { 1 } { r + p }$

$\Rightarrow$ $\frac { p + q - r - p } { ( r + p ) ( p + q ) } = \frac { r + p - q - r } { ( q + r ) ( r + p ) }$

$\Rightarrow$ $\frac { q - r } { p + q } = \frac { p - q } { q + r }$ or $q ^ { 2 } - r ^ { 2 } = p ^ { 2 } - q ^ { 2 }$

$\therefore$ $2 q ^ { 2 } = r ^ { 2 } + p ^ { 2 }$

Therefore $p ^ { 2 } , q ^ { 2 } , r ^ { 2 }$ are in A.P.