The locus of the centre of a circle touching the lines
\[
x+2y=0
\]
and
\[
x-2y=0
\]
is
Show Hint
The centre of a circle touching two intersecting lines always lies on one of their angle bisectors.
Use
\[
\boxed{\frac{L_1}{\sqrt{a_1^2+b_1^2}}
=
\pm
\frac{L_2}{\sqrt{a_2^2+b_2^2}}}
\]
to find the angle bisectors.