Question

if a = (√5+1) /(√5-1) and ab = 1, then find the value of (a/b + b/a) ​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\:a = \dfrac{ \sqrt{5} + 1}{ \sqrt{5} - 1}$

So, On rationalizing the denominator, we get

$\rm :\longmapsto\:a = \dfrac{ \sqrt{5} + 1}{ \sqrt{5} - 1} \times \dfrac{ \sqrt{5} + 1}{ \sqrt{5} + 1}$

We know,

$\boxed{\tt{ (x + y)(x - y) = {x}^{2} - {y}^{2}}}$

So, using this, we get

$\rm :\longmapsto\:a = \dfrac{ {( \sqrt{5} + 1)}^{2} }{ {( \sqrt{5} )}^{2} - {1}^{2} }$

$\rm :\longmapsto\:a = \dfrac{5 + 1 + 2 \sqrt{5} }{5 - 1}$

$\rm :\longmapsto\:a = \dfrac{6 + 2 \sqrt{5} }{5 - 1}$

$\rm :\longmapsto\:a = \dfrac{2(3 + \sqrt{5} )}{4}$

$\rm :\longmapsto\:a = \dfrac{3 + \sqrt{5}}{2}$

Now, Further given that

$\rm :\longmapsto\:ab = 1$

$\bf\implies \:b = \dfrac{1}{a}$

$\bf\implies \:b = \dfrac{2}{3 + \sqrt{5} }$

So, on rationalizing the denominator, we get

$\rm :\longmapsto\:b = \dfrac{2}{3 + \sqrt{5} } \times \dfrac{3 - \sqrt{5} }{3 - \sqrt{5} }$

$\rm :\longmapsto\:b = \dfrac{2(3 - \sqrt{5}) }{ {3}^{2} - {( \sqrt{5} )}^{2} }$

$\rm :\longmapsto\:b = \dfrac{2(3 - \sqrt{5}) }{ 9 - 5}$

$\rm :\longmapsto\:b = \dfrac{2(3 - \sqrt{5}) }{4}$

$\rm :\longmapsto\:b = \dfrac{3 - \sqrt{5} }{2}$

Now, we have,

$\rm :\longmapsto\:b = \dfrac{3 - \sqrt{5} }{2}$

and

$\rm :\longmapsto\:a = \dfrac{3 + \sqrt{5} }{2}$

Now, Consider

$\rm :\longmapsto\:\dfrac{a}{b} + \dfrac{b}{a}$

$\rm \: = \: \dfrac{ {a}^{2} + {b}^{2} }{ab}$

$\rm \: = \: \dfrac{ {a}^{2} + {b}^{2} }{1}$

$\rm \: = \: {a}^{2} + {b}^{2}$

$\rm \: = \: {\bigg[\dfrac{3 + \sqrt{5} }{2} \bigg]}^{2} + {\bigg[\dfrac{3 - \sqrt{5} }{2} \bigg]}^{2}$

$\rm \: = \: \dfrac{1}{4}\bigg[ {(3 + \sqrt{5}) }^{2} + {(3 - \sqrt{5}) }^{2} \bigg]$

$\rm \: = \: \dfrac{1}{4}\bigg[ 9 + 5 + 6 \sqrt{5} + 9 + 5 - 6 \sqrt{5} \bigg]$

$\rm \: = \: \dfrac{1}{4}\bigg[28\bigg]$

$\rm \: = \: 7$

Hence,

$\rm :\longmapsto\:\boxed{\tt{ \: \dfrac{a}{b} + \dfrac{b}{a} = 7 \: }}$

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FORMULA USED

$\boxed{\tt{ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab}}$

$\boxed{\tt{ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab}}$

$\boxed{\tt{ (a + b)(a - b) = {a}^{2} - {b}^{2}}}$

Answer 2

Answer:

Hi! can u pls solve this for me:)

as ur a genius like Einstein

Pls solve this^^

I know I m asking this is ur question.....

but... pls

I M NEEDING THIS URGENT SO ...