Question

$if \: \alpha \: and \: \beta \: are \: the \: roots \: of \: eqaution \: {y}^{2} - 5y + 9 = 0 \: then \: value \: of \: \sqrt{ \frac{ \alpha }{ \beta } } \: + \sqrt{ \frac{ \beta }{ \alpha } } is$
•0
•5/9
•5/3
•9/5​

Answer 1

Answer:

5/3

Step-by-step explanation:

${y}^{2} - 5y + 9 = 0 \\ = > y = \frac{5± \sqrt{ {5}^{2} - 4 \times 9 \times 1} }{2} \\ = > y = \frac{5± \sqrt{ - 11} }{2} \\ = > y = \frac{5±i \sqrt{11} }{2} \\ = > \frac{5 + i \sqrt{11} }{2} \: or \: \frac{5 - i \sqrt{11} }{2}$

now putting those values in

$\sqrt{ \frac{ \alpha }{ \beta } } + \sqrt{ \frac{ \beta }{ \alpha } } \\ = \sqrt{ \frac{ \frac{5 + i \sqrt{11} }{2} }{ \frac{5 - i \sqrt{11} }{2} } } + \sqrt{ \frac{ \frac{5 - i \sqrt{11} }{2} }{ \frac{5 + i \sqrt{11} }{2} } } \\ = \sqrt{ \frac{5 + i \sqrt{11} }{5 - i \sqrt{11} } } + \sqrt{ \frac{5 - i \sqrt{11} }{5 + i \sqrt{11} } } \\ = \frac{ { (\sqrt{5 + i \sqrt{11} }) }^{2} + {( \sqrt{5 - i \sqrt{11} } )}^{2} }{ \sqrt{(5 + i \sqrt{11})(5 - i \sqrt{11}) } } \\ = \frac{5 + i \sqrt{11} + 5 - i \sqrt{11} }{ \sqrt{ {5}^{2} - {(i \sqrt{11}) }^{2} } } \\ = \frac{10}{ \sqrt{25 + 11} } \\ = \frac{10}{6} \\ = \frac{5}{3}$

Hope it helps :D