Question

2) If a + 1/a = m and a - 1/a = n, find the value of
m²- n²
​

Answer 1

m = a + 1/a

therefore m^2 = ( a + 1/a)^2

{ (a + b)^2 = a^2 + b^2 + 2ab } and {(a - b)^2 = a^2 + b^2 - 2ab}

so,

m^2 = a^2 + 1/a^2 + 2

n^2 = a^2 + 1/a^2 - 2

therefore m^2 - n^2 = (a^2 + 1/a^2 + 2) -(a^2 + 1/a^2 -2)

= a^2 + 1/a^2 + 2 -a^2 - 1/a^2 + 2

= 4

Answer 2

Answer :

m² - n² = 4

Solution :

Given :

a + 1/a = m -------(1)

a - 1/a = n -------(2)

To find :

m² - n² = ?

Now ,

Adding eq-(1) and (2) , we get ;

=> a + 1/a + a - 1/a = m + n

=> 2a = m + n

=> m + n = 2a ---------(3)

Now ,

Subtracting eq-(2) from (1) , we get ;

=> (a + 1/a) - (a - 1/a) = m - n

=> a + 1/a - a + 1/a = m - n

=> 2/a = m - n

=> m - n = 2/a ---------(4)

Now ,

Multiplying eq-(3) and (4) , we get ;

=> (m + n) × (m - n) = 2a × 2/a

=> m² - n² = 2 × 2

=> m² - n² = 4

Hence ,

m² - n² = 4