The equations of the tangents to the circles $x^2+y^2=a^2$ which makes a triangle  with axes  of area $a^2$ is

Equation of a tangent to $x^2+y^2=a^2$ is $y=mx\pm a\sqrt{1+m^2}$ Area of the triangle formed by this line with the coordinate axes is $a^2\Rightarrow \frac{a^2\left(1+m^2\right)}{2\left|m\right(-1\left)\right|}=a^2\Rightarrow 1+m^2=2|-m|\Rightarrow m^2+1=\pm 2m\Rightarrow (m\pm 1{)}^2=0\Rightarrow m=\pm 1$
Required tangents are $y=\pm x\pm a\sqrt{2}$.