Question-78108

Question Number 78108 by mr W last updated on 14/Jan/20

Commented by mr W last updated on 14/Jan/20

f i n d t h e r e l a t i o n b e t w e e n a a n d c .

(i r e p o s t e d t h e q u e s t i o n h e r e t o a v o i d

t h a t t h e s o l u t i o n w i l l b e d e l e t e d b y

s o m e o n e .)

Answered by mr W last updated on 14/Jan/20

(i) + (i i) :

32 (\frac{c}{a} + \frac{a}{c}) = {(\frac{x}{a})}^{5} + 5 {(\frac{x}{a})}^{3} + 10 (\frac{x}{a}) + 10 (\frac{a}{x}) + 5 {(\frac{a}{x})}^{3} + {(\frac{a}{x})}^{5}

32 (\frac{c}{a} + \frac{a}{c}) = {(\frac{x}{a} + \frac{a}{x})}^{5}

\Rightarrow \frac{x}{a} + \frac{a}{x} = 2 {(\frac{c}{a} + \frac{a}{c})}^{\frac{1}{5}} \dots (I)

(i) - (i i) :

32 (\frac{c}{a} - \frac{a}{c}) = {(\frac{x}{a})}^{5} - 5 {(\frac{x}{a})}^{3} + 10 (\frac{x}{a}) - 10 (\frac{a}{x}) + 5 {(\frac{a}{x})}^{3} - {(\frac{a}{x})}^{5}

32 (\frac{c}{a} - \frac{a}{c}) = {(\frac{x}{a} - \frac{a}{x})}^{5}

\Rightarrow \frac{x}{a} - \frac{a}{x} = 2 {(\frac{c}{a} - \frac{a}{c})}^{\frac{1}{5}} \dots (I I)

{(I)}^{2} - {(I I)}^{2} :

4 = 4 {(\frac{c}{a} + \frac{a}{c})}^{\frac{2}{5}} - 4 {(\frac{c}{a} - \frac{a}{c})}^{\frac{2}{5}}

\Rightarrow {(\frac{c}{a} + \frac{a}{c})}^{\frac{2}{5}} - {(\frac{c}{a} - \frac{a}{c})}^{\frac{2}{5}} = 1

o r

\Rightarrow \sqrt[5]{{(\frac{c}{a})}^{2} + {(\frac{a}{c})}^{2} + 2} - \sqrt[5]{{(\frac{c}{a})}^{2} + {(\frac{a}{c})}^{2} - 2} = 1

l e t {(\frac{c}{a})}^{2} + {(\frac{a}{c})}^{2} - 2 = s

\Rightarrow \sqrt[5]{s + 4} - \sqrt[5]{s} = 1

\Rightarrow \sqrt[5]{s + 4} = \sqrt[5]{s} + 1

\Rightarrow s + 4 = s + 5 {(\sqrt[5]{s})}^{4} + 10 {(\sqrt[5]{s})}^{3} + 10 {(\sqrt[5]{s})}^{2} + 5 (\sqrt[5]{s}) + 1

\Rightarrow\Rightarrow 5 {(\sqrt[5]{s})}^{4} + 10 {(\sqrt[5]{s})}^{3} + 10 {(\sqrt[5]{s})}^{2} + 5 (\sqrt[5]{s}) - 3 = 0

l e t t = \sqrt[5]{s} = \sqrt[5]{{(\frac{c}{a})}^{2} + {(\frac{a}{c})}^{2} - 2}

\Rightarrow 5 t^{4} + 10 t^{3} + 10 t^{2} + 5 t - 3 = 0

\Rightarrow 5 {(t^{2} + t)}^{2} + 5 (t^{2} + t) - 3 = 0

\Rightarrow t^{2} {(t + 1)}^{2} + t (t + 1) - \frac{3}{5} = 0

\Rightarrow t^{2} {(t + 1)}^{2} + t (t + 1) + \frac{1}{4} - (\frac{3}{5} + \frac{1}{4}) = 0

\Rightarrow {[t (t + 1) + \frac{1}{2}]}^{2} = {(\frac{\sqrt{85}}{10})}^{2}

\Rightarrow t (t + 1) + \frac{1}{2} = \frac{\sqrt{85}}{10}

\Rightarrow t^{2} + t + \frac{1}{2} - \frac{\sqrt{85}}{10} = 0

\Rightarrow t = \frac{1}{4} (- 1 \pm \sqrt{\frac{8 \sqrt{85}}{5} - 7})

\Rightarrow \sqrt[5]{{(\frac{c}{a})}^{2} + {(\frac{a}{c})}^{2} - 2} = \frac{1}{4} (- 1 + \sqrt{\frac{8 \sqrt{85}}{5} - 7})

\Rightarrow \frac{c}{a} - \frac{a}{c} = \pm \frac{1}{2^{5}} {(- 1 + \sqrt{\frac{8 \sqrt{85}}{5} - 7})}^{5 / 2} = μ

{(\frac{c}{a})}^{2} - μ (\frac{c}{a}) - 1 = 0

\Rightarrow \frac{c}{a} = \frac{1}{2} (μ \pm \sqrt{μ^{2} + 4})

\Rightarrow \frac{c}{a} = \pm \frac{1}{64} {{(- 1 + \sqrt{\frac{8 \sqrt{85}}{5} - 7})}^{5 / 2} \pm \sqrt{{(- 1 + \sqrt{\frac{8 \sqrt{85}}{5} - 7})}^{5} + 4096}}

o r

\frac{c}{a} \approx \pm 0.971516, \pm 1.029319

Commented by TawaTawa last updated on 14/Jan/20

Weldone sir . God bless you .