It is given that $f(x) = x$, a circle is enclosed by $y = |f(x)| - 4$ and \(y = 4 - |f(x)|\

Question

It is given that $f(x) = x$, a circle is enclosed by $y = |f(x)| - 4$ and $y = 4 - |f(x)|$. Find the equation of the circle. The line $y = x$ meets the circle at $A(a, a)$, where $a < 0$. Find the coordinates of $A$. Given that $N$ is the lowest point on the circle, which is then reflected in the line $y = x$ to point $M$, find the difference between the areas of the circle and triangle $AMN$, leaving your answer in the exact form.

Accepted Answer

(i) ${{x}^{2}}+{{y}^{2}}=8$ (ii) $A\left( -2,-2 \right)$ (iii) $\left( 8\pi -4\sqrt{2}+4 \right)$ units$^{2}$