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Question

# If α, β are roots of the equation ${x}^{2}+lx+m=0$, write an equation whose roots are $-\frac{1}{\alpha }\mathrm{and}-\frac{1}{\beta }$.

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Solution

## Given equation: ${x}^{2}+lx+m=0$ Also, $\alpha and\beta$ are the roots of the equation. Sum of the roots = $\alpha +\beta =\frac{-l}{1}=-l$ Product of the roots = $\alpha \beta =\frac{m}{1}=m$ Now, sum of the roots = $-\frac{1}{\alpha }-\frac{1}{\beta }=-\frac{\alpha +\beta }{\alpha \beta }=-\frac{-l}{m}=\frac{l}{m}$ Product of the roots = $\frac{1}{\alpha \beta }=\frac{1}{m}$ $\therefore {x}^{2}-\left(S\mathrm{um}\mathrm{of}the\mathrm{roots}\right)x+P\mathrm{roduct}\mathrm{of}the\mathrm{roots}=0\phantom{\rule{0ex}{0ex}}⇒{x}^{2}-\frac{l}{m}x+\frac{1}{m}=0\phantom{\rule{0ex}{0ex}}⇒m{x}^{2}-lx+1=0$ Hence, this is the equation whose roots are $-\frac{1}{\alpha }and-\frac{1}{\beta }.$

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