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  4. If α and β are the roots of the equation x2-7x+1=0, then the value of (1/(α -7)2)+(1/(β -7)2) is:

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Q. If $ \alpha $ and $ \beta $ are the roots of the equation $ {{x}^{2}}-7x+1=0, $ then the value of $ \frac{1}{{{(\alpha -7)}^{2}}}+\frac{1}{{{(\beta -7)}^{2}}} $ is:

KEAMKEAM 2006

Solution:

Given $ \alpha $ and $ \beta $ are the roots of $ {{x}^{2}}-7x+1=0 $ $ \Rightarrow $ $ \alpha +\beta =7 $ and $ \alpha \beta =1 $ $ \therefore $ $ \alpha -7=\beta ,\beta -7=-\alpha $ $ \therefore $ $ \frac{1}{{{(\alpha -7)}^{2}}}+\frac{1}{{{(\beta -7)}^{2}}}=\frac{1}{{{\beta }^{2}}}+\frac{1}{{{\alpha }^{2}}} $ $ =\frac{{{\alpha }^{2}}+{{\beta }^{2}}}{(\alpha {{\beta }^{2}})} $ $ ={{(\alpha +\beta )}^{2}}-2\alpha \beta $ $ =49-2=47 $