Given-x-y-R-and-x-5-y-3-5-x-3-y-139-If-maximum-and-minimum-of-x-y-xy-is-M-and-n-respectively-then-what-the-value-of-3M-4n-

Question Number 158259 by cortano last updated on 01/Nov/21

G i v e n x, y \in R^{+} a n d (\frac{x}{5} + \frac{y}{3}) (\frac{5}{x} + \frac{3}{y}) = 139 .

I f m a x i m u m a n d m i n i m u m

o f \frac{x + y}{\sqrt{x y}} i s M a n d n r e s p e c t i v e l y,

t h e n w h a t t h e v a l u e o f 3 M - 4 n .

Commented by tounghoungko last updated on 01/Nov/21

\Leftrightarrow (\frac{x}{5} + \frac{y}{3}) (\frac{5}{x} + \frac{3}{y}) = 139

\Leftrightarrow \frac{3 x}{5 y} + \frac{5 y}{3 x} = 137 \to {\begin{cases} \frac{x}{y} = a \\ \frac{y}{x} = \frac{1}{a} \end{cases}

\Leftrightarrow \frac{3 a}{5} + \frac{5}{3 a} = 137 \dots (i)

\Rightarrow \frac{x + y}{\sqrt{x y}} = \sqrt{\frac{x}{y}} + \sqrt{\frac{y}{x}} = \sqrt{a} + \frac{1}{\sqrt{a}}

\Rightarrow f r o m (i) \Rightarrow 9 a^{2} - 2055 a + 25 = 0

b y {V i e t h a}^{'} s r u l e {\begin{cases} a_{1} + a_{2} = \frac{2055}{9} \\ a_{1} \times a_{2} = \frac{25}{9} \end{cases}

l e t {\begin{cases} m a x = M = \sqrt{a_{1}} + \frac{1}{\sqrt{a_{1}}} \\ m i n = n = \sqrt{a_{2}} + \frac{1}{\sqrt{a_{2}}} \end{cases}

{(3 M - 4 n)}^{2} = {(3 \sqrt{a_{1}} + \frac{3}{\sqrt{a_{1}}} - 4 \sqrt{a_{2}} - \frac{4}{\sqrt{a_{2}}})}^{2}

= {(\frac{3 a_{1} + 3}{\sqrt{a_{1}}} - 4 (\frac{a_{2} + 1}{\sqrt{a_{2}}}))}^{2}

= {(\frac{3 a_{1} \sqrt{a_{2}} + 3 \sqrt{a_{2}} - 4 a_{2} \sqrt{a_{1}} - 4 \sqrt{a_{1}}}{\sqrt{a_{1} a_{2}}})}^{2}

= \frac{{(\sqrt{a_{1} a_{2}} (3 \sqrt{a_{1}} - 4 \sqrt{a_{2}}) + 3 \sqrt{a_{2}} - 4 \sqrt{a_{1}})}^{2}}{\frac{25}{9}}

= \frac{9}{25} (a_{1} a_{2} (9 a_{1} + 16 a_{2} - 12 \sqrt{a_{1} a_{2}}) + 9 a_{2} + 16 a_{1} - 12 \sqrt{a_{1} a_{2}} + 2 \sqrt{a_{1} a_{2}} (9 \sqrt{a_{1} a_{2}} - 12 a_{1} - 12 a_{2} + 16 \sqrt{a_{1} a_{2}})

Answered by mr W last updated on 01/Nov/21

(\frac{x}{5} + \frac{y}{3}) (\frac{5}{x} + \frac{3}{y}) = 139

\frac{5}{3} (\frac{y}{x}) + \frac{3}{5} (\frac{x}{y}) + 2 = 139

l e t t = \frac{y}{x}

25 t + \frac{9}{t} = 137 \times 15

25 t^{2} - 2055 t + 9 = 0

t = \frac{3 (137 \pm 3 \sqrt{2085})}{10}

P = {(\frac{x + y}{\sqrt{x y}})}^{2} = \frac{x}{y} + \frac{y}{x} + 2 = t + \frac{1}{t} + 2

P_{1} = \frac{3 (137 + 3 \sqrt{2085})}{10} + \frac{10}{3 (137 + 3 \sqrt{2085})} + 2

= \frac{2359 - 24 \sqrt{2085}}{15} = n^{2}

P_{2} = \frac{3 (137 - 3 \sqrt{2085})}{10} + \frac{10}{3 (137 - 3 \sqrt{2085})} + 2

= \frac{2359 + 24 \sqrt{2085}}{15} = M^{2}

3 M - 4 n = 3 \sqrt{\frac{2359 + 24 \sqrt{2085}}{15}} - 4 \sqrt{\frac{2359 - 24 \sqrt{2085}}{15}} \approx 8.962

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