Question

If * is defined on the set R of all real numbers by $a\ast b=\sqrt{a^2+b^2}$, find the identity element in R with respect to *.

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is A 0

Let e be the identity element in R with respect to *. Then,

$a\ast e=a=e\ast a$ for all $a\in R$

$a\ast e=a$ and $e\ast a=a$ for all $a\in R$

$\sqrt{a^2+e^2}=a$ and $\sqrt{e^2+a^2}=a$ for all $a\in R$

$a^2+e^2=a^2$ and $e^2+a^2=a^2$ for all $a\in R$

$e=0$

Hence, 0 is the identity element in R with respect to $\ast$.