• using integeration find the area of the region

{(x,y): x² + y²

• show that the curves xy= a² and x²+y²= 2a² touch each other

The equation of the given curves are $xy=a^2 and x^2+y^2=2a^2$
solving these two equations we have
 $$\begin{gathered} x^2+(\frac{a^2}{x}{)}^2=2a^2 \\ {x}^4-2a^2{x}^2+a^4=0 \\ (x^2-a^2)=0 \\ \Rightarrow x^2=a^2 \\ \Rightarrow x=\pm a \end{gathered}$$
and $y=\pm a$
therefore the common points to the given curves are (a,a) and (-a,-a)
the slopes of the two curves at each  point in common is -1.
hence the angle of intersection of the curves is ${\tan }^{-1}0=0$.
therefore the given curves touches at the common point.

hope this helps you.