Find the value of r in the following expression.
$$(J - p - r - s)/q + (J - p - q - s)/r + (J - q - r - s)/p + (J - p - q - r)/s = 4$$
$J, p, q, r, s$ are real and $p + q + s = 3$.
Find the value of $r$.
Find the value of r in the following expression.
$$(J - p - r - s)/q + (J - p - q - s)/r + (J - q - r - s)/p + (J - p - q - r)/s = 4$$
$J, p, q, r, s$ are real and $p + q + s = 3$.
Find the value of $r$.
Note $$(J - p - r - s)/q =(J+q-3-r)/q$$ $$(J - p - q - s)/r =(J-3)/r$$$$(J - q - r - s)/p=(J+p-3-r)/p$$$$ (J - p - q - r)/s = (J+s-3-r)/s$$
Summing them up we get $$ (J-3)(\frac{1}{p}+\frac{1}{q}+\frac{1}{s}+\frac{1}{r})-r(\frac{1}{p}+\frac{1}{q}+\frac{1}{s})+3=4$$ Assuming $$\frac{1}{p}+\frac{1}{q}+\frac{1}{s}=A$$And$$(J-3)=B$$ We get $$ B(A+\frac{1}{r})-rA-1=0$$ Rearranging we get the following quadratic$$Ar^2-rAB-B+r=0$$$$(rA+1)(r-B)=0$$ $$r=-\frac{1}{A}=\frac{-pqs}{ps+pq+qs}$$$$r=B=(J-3)$$
Also $r=\frac{-qp(-3+p+q)}{(qp-3p+p^2-3q+q^2)}$ (In addition to $J-3$)