Question

The number of pairs $(a,b)$ of positive real numbers satisfying $a^4+b^4<1$ and $a^2+b^2>1$ is/are:

• A. $0$
• B. $1$
• C. $2$
• D. more than $2$

Answer 1

The correct option is D more than $2$

Let $a^2=x\text{and}{b}^2=y$, then the given equation can be written as,
$x^2+y^2<1x+y>1$



From the given condition $(x,y)$ should lie inside the circle $x^2+y^2=1$ and above the line $x+y=1$ which is the shaded region.
The number of pairs of $(x,y)$ can be infinite, so the number of pairs of$(a,b)$ will also be infinite.