Question

value of x,y.where x<1​

Answer 1

Answer :-

x = 1/2, y = 4

Solution :-

Given equation :-

$\implies \sf \dfrac{2}{3} = \bigg(x - \dfrac{1}{y} \bigg) + \bigg( {x}^{2} - \dfrac{1}{ {y}^{2} } \bigg)...... \infty$

On simplifying the equation,

$\implies \sf \dfrac{2}{3} = \bigg( x + {x}^{2} + {x}^{3} ........ \infty \bigg)- \bigg( \dfrac{1}{y} + \dfrac{1}{ {y}^{2} } + \dfrac{1}{ {y}^{3} } ...... \infty \bigg)$

If we look the terms in RHS it looks like infinite GP

Here, we have 2 Infinite GPs

Finding common ratio

The Common ratios of GP are x and 1/y

Given

• xy = 2
• x < 1

Find the value of y in terms of x

Given, xy = 2

==> x = 2/y

Given, x < 1

==> 2/y < 1

==> 2 < y

==> y > 2

==> 1/2 > y

==> y < 1/2

Since, the Common ratio of both infinite GPs is r < 1

Hence, we can use the below Sum of infinite terms of GP formula

$\sf S = \dfrac{a}{1 - r}$

$\implies \sf \dfrac{2}{3} = \bigg( x + {x}^{2} + {x}^{3} ........ \infty \bigg)- \bigg( \dfrac{1}{y} + \dfrac{1}{ {y}^{2} } + \dfrac{1}{ {y}^{3} } ...... \infty \bigg)$

Here,

In 1st infinite GP :

• a = x
• r = x

In 2nd infinite GP :

• a = 1/y
• r = 1/y

Substituting the values in formula

$\implies \sf \dfrac{2}{3} = \dfrac{x}{1 - x} - \dfrac{ \dfrac{1}{y} }{1 - \dfrac{1}{y} }$

$\implies \sf \dfrac{2}{3} = \dfrac{x}{1 - x} - \dfrac{ \dfrac{1}{y} }{ \dfrac{y - 1}{y} }$

$\implies \sf \dfrac{2}{3} = \dfrac{x}{1 - x} - \dfrac{1}{y - 1}$

Taking LCM

$\implies \sf \dfrac{2}{3} = \dfrac{x(y - 1) - 1(1 - x)}{(1 - x)(y - 1)}$

$\implies \sf \dfrac{2}{3} = \dfrac{xy - x - 1 + x}{1 (y - 1)- x(y - 1)}$

$\implies \sf \dfrac{2}{3} = \dfrac{xy - 1}{y - 1- xy + x}$

$\implies \sf \dfrac{2}{3} = \dfrac{2- 1}{y - 1- 2 + x}$

$\implies \sf \dfrac{2}{3} = \dfrac{1}{x + y - 3 }$

==> 2x + 2y - 6 = 3

==> 2x + 2( 2/x ) - 6 - 3 = 0

[ Since y = 2/x ]

==> 2x + 4/x - 9 = 0

==> 2x² - 9x + 4 = 0

==> 2x² - 8x - x + 4 = 0

==> 2x(x - 4) - 1(x - 4) = 0

==> (2x - 1)(x - 4) = 0

==> 2x - 1 = 0 or x - 4 = 0

==> 2x = 1 or x = 4

==> x = 1/2 or x = 4

Since x < 1

==> x = 1/2

When x = 1/2

==> y = 2/x = 2/ ( 1/2 ) = 4

Therefore the value of x and y are 1/2, 4, respectively.