Question

If $M=\frac{4pq}{3p+q}$ then find the value of $\frac{M+4p}{M-p}+\frac{M-3p}{M-q}$

Answer 1

$\Rightarrow \frac{\frac{4pq}{3p+q}+4p}{\frac{4pq}{3p+q}-p}+\frac{\frac{4pq}{3p+q}-3p}{\frac{4pq}{3p+q}-q}$

$\Rightarrow \frac{4pq+4p(3p+q)}{4pq-p(3p+q)}+\frac{4pq-3p(3p+q)}{4pq-q(3p+q)}$

$\Rightarrow \frac{4pq+4p\times 3p+4pq}{4pq-3p^2+pq}+\frac{4pq-9p^2-3pq}{4pq-3pq-q^2}$

$\Rightarrow \frac{8pq+12p^2}{5pq-3p^2}+\frac{pq-9p^2}{pq-q^2}$

$\Rightarrow \frac{8q+12p}{5q-3p}+\frac{pq-9p^2}{pq-q^2}$

$\Rightarrow \frac{(8q+12p)(pq-q^2)+(pq-9p^2)(5q-3p)}{(5q-3p)(pq-q^2)}$

$\Rightarrow \frac{8p{q}^2-8q^3+12p^{2}q-12p{q}^2+5p{q}^2-3p^{2}q-45p^{2}q+27p^3}{5p{q}^2-5q^3-3p^{2}q+3p{q}^2}$

$\Rightarrow \frac{27p^3-8q^3+p{q}^2-37p^{2}q}{8p{q}^2-5q^3-3p^{2}q}$.