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Question Number 69018 by Maclaurin Stickker last updated on 17/Sep/19

Commented by Prithwish sen last updated on 18/Sep/19

$$\mathrm{It}\mathrm{is}\mathrm{only}\mathrm{possible}\mathrm{when}$$$$4\mathbf{a}+5\mathbf{b}+1=8\mathbf{ab}+1$$$$\Rightarrow 4\mathbf{a}+5\mathbf{b}=8\mathbf{ab}............\left(\mathbf{i}\right)$$$$\mathbf{and}$$$$(16{\mathbf{a}}^2+{\mathbf{b}}^2+1)(4\mathbf{a}+5\mathbf{b}+1)={(4\mathbf{a}+5\mathbf{b}+1)}^2=({(8\mathbf{ab}+1)}^2......\left(\mathbf{ii}\right)$$$$\Rightarrow {16\mathrm{a}}^2+{\mathrm{b}}^2+1=4\mathrm{a}+5\mathrm{b}+1$$$${16\mathrm{a}}^2+{\mathrm{b}}^2+1=8\mathrm{ab}+1\mathbf{from}\left(\mathbf{i}\right)$$$${(4\mathrm{a}-\mathrm{b})}^2=0$$$$\Rightarrow 4\mathbf{a}=\mathbf{b}.......\left(\mathbf{ii}\right)$$$$\therefore \mathrm{from}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\mathrm{we}\mathrm{get}$$$$\mathbf{a}=0,\mathbf{b}=0\mathbf{and}\mathbf{a}=\frac{3}{4}\mathbf{and}\mathbf{b}=3$$$$\because \mathbf{the}\mathbf{equation}\mathbf{only}\mathbf{satified}\mathbf{by}$$$$\mathbf{a}=\frac{3}{4}\mathbf{and}\mathbf{b}=3$$$$\therefore \mathbf{the}\mathbf{possible}\mathbf{solution}\mathbf{is}(\frac{3}{4},3)$$

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