Question

If the ordered pairs(x1,y1) and (x2,y2) satisfying the system of equations log10(y)-log10|x| = log100(4) and log100| x + y |=1/2 find value of x1x2y1y2

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ {\log }_{10}\mathrm{y}-{\log }_{10}\left|\mathrm{x}\right|={\log }_{100}4 \\ {\log }_{10}\left(\frac{\mathrm{y}}{\left|\mathrm{x}\right|}\right)={\log }_{{10}^2}4=\frac{1}{2}{\log }_{10}4={\log }_{10}\sqrt{4} \\ \frac{\mathrm{y}}{\left|\mathrm{x}\right|}=2\Rightarrow \mathrm{y}=2\mathrm{x}\mathrm{or}\mathrm{y}=-2\mathrm{x}.....\left(1\right) \\ \mathrm{Again} \\ {\log }_{100}\left|\mathrm{x}+\mathrm{y}\right|=1/2\Rightarrow \left|\mathrm{x}+\mathrm{y}\right|=10 \\ \mathrm{x}+\mathrm{y}=10........\left(2\right)\left[\mathrm{Neglecting}\mathrm{x}+\mathrm{y}=-10\mathrm{because}\log \mathrm{cannot}\mathrm{take}\mathrm{negative}\mathrm{value}\right] \\ \mathrm{Now}\mathrm{solnving}\mathrm{y}=2\mathrm{x}\mathrm{and}\mathrm{x}+\mathrm{y}=10 \\ 3\mathrm{x}=10\Rightarrow \mathrm{x}=\frac{10}{3} \\ \mathrm{Now}\mathrm{basically}\mathrm{th}isis{\mathrm{x}}_1=\frac{10}{3} \\ \mathrm{So}\mathrm{y}=\frac{20}{3}\mathrm{and}\mathrm{th}isis{\mathrm{y}}_1 \\ \mathrm{Now}\mathrm{solnving}\mathrm{y}=-2\mathrm{x}\mathrm{and}\mathrm{x}+\mathrm{y}=10 \\ -\mathrm{x}=-10\Rightarrow \mathrm{x}=10 \\ \mathrm{Now}\mathrm{basically}\mathrm{this}\mathrm{is}{x}_2=10. \\ \mathrm{So}\mathrm{y}=-20\mathrm{and}\mathrm{this}is{y}_2 \\ \mathrm{So}{\mathbf{x}}_{\mathbf{1}}\mathbf{\times }\mathbf{}{\mathbf{x}}_{\mathbf{2}}\mathbf{\times }\mathbf{}{\mathbf{y}}_{\mathbf{1}}\mathbf{\times }{\mathbf{y}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{10}}{\mathbf{3}}\mathbf{\times }\frac{\mathbf{20}}{\mathbf{3}}\mathbf{\times }\mathbf{10}\mathbf{\times }\mathbf{-}\mathbf{20}\mathbf{=}\frac{\mathbf{-}\mathbf{40000}}{\mathbf{9}} \end{gathered}$$
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