For the least possible value of n Z the solution (x, y) of the equations ⁻¹ x+ ( ⁻¹ y )^2= n ^2 4 and ⁻¹ x ( ⁻¹ y )^2= ^4 16 , is

Step-by-step solution

( ⁻¹ y )^2+ ⁻¹ x= n ^2 4 and ( ⁻¹ x ) ( ⁻¹ y )^2= ^4 16 Let ⁻¹ x=a and ⁻¹ y=b a+b^2= n ^2 4 (i)a b^2= ^4 16 (ii) a-b= (a+b^2 )^2-4 a b = n^2 ^4 16 - 4 ^4 16 Now, least value of n can be 2 a-b^2= 4 ^4 16 - 4 ^4 16 a-b^2=0 (iii) Adding Eqs. (i) and (iii), we get 2 a= n ^2 4 or 2 a= 2 ^2 4 ( n=2) or a= ^2 4 ⁻¹ x= ^2 4 = ( ^2 4 ) Now, from Eq. (iii), a=b ( ⁻¹ y )^2= ^2 4 or ⁻¹ y= 2 or y= ( 2 ) or y= 1 Therefore, ordered pair is ( ^2 4 , 1 ) .