<s>[INST] $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6,\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=8$,then find $\frac{a}{b}+\frac{c}{d}$ $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6,\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=8$,then find $\frac{a}{b}+\frac{c}{d}$

I have tried multiplying the two given equations $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6,\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=8$ but not helpful then i tried adding the two equations $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6,\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=8$ but this is also not helpful to find $\frac{a}{b}+\frac{c}{d}$.
Neither $AM-GM$ inequality method is helpful.
Please suggest me a way to find $\frac{a}{b}+\frac{c}{d}$.
 [/INST] Hint:
Let $$\alpha=\frac ab+\frac cd,\beta=\frac bc +\frac da$$
Consider
$$\alpha\beta=(\frac ab+\frac cd)(\frac bc +\frac da)=\frac ac+\frac bd+\frac db+\frac ca=8$$
And note that
$$\alpha+\beta=6$$
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