Question

please provide a solution

Answer 1

$\dfrac{ p^{2} + q^{2}}{r^{2}+s^{2}} = \dfrac{pq}{rs}$


Multiply and divide by 2 in right hand side,


$\dfrac{ p^{2} + q^{2}}{r^{2}+s^{2}} = \dfrac{2pq}{2rs} \\ \\ \\ \dfrac{ p^{2} + q^{2}}{2pq} = \dfrac{r^{2}+s^{2}}{2rs}$



By Componendo and dividendo,



$\dfrac{ p^{2} + q^{2} + 2pq}{p^{2}+q^{2}-2pq} = \dfrac{r^{2}+s^{2}+2rs}{r^{2}+s^{2}-2rs}$


$\dfrac{ ( p + q)^{2}}{(p-q)^{2}} = \dfrac{(r+s)^{2}}{(r-s)^{2}}$



Square root on both sides,


$\dfrac{ p+q}{p-q}=\dfrac{r+s}{r-s}$


$\dfrac{r-s}{r+s}=\dfrac{p-q}{p+q}$





Therefore the value of ( p - q ) / ( p + q ) in terms of r and s is ( r - s ) / ( r + s )



$\mathsf{Option \: 2 \: is \: correct.} \:$