Question

If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .

Answer 1

Let * be an operation defined on R 

a * b = `sqrt(a^2 + b^2 ) `

Let ‘e’ be the identity element

then a*e = e*a = a

`sqrt(a^2 + e^2 ) = sqrt(e^2 + a^2 ) = a `

B squaring on both side

e² + a² = a² 

e² = 0 ⇒ e = 0 

a*0 = `sqrt( a^2 + 0^2 ) ` = |a| ≠ a

i.e. if a = -2 (or any other negative real no.)

then a* 0 ≠ -2 

since a*0 = `sqrt(a^2) `= | a| = 2 ≠ - 2

Hence identity element does not exists.