Find the equation of the locus of the intersection of the lines below Find the equation of the locus of the intersection of the lines below
$y=mx+\sqrt{m^2+2}$
$y=-\frac{ 1 }{ m }x+\sqrt{\frac{ 1 }{ m^2 +2}}$

By graphing, I have got an ellipse as locus : $x^2+\dfrac{y^2}{2}=1$.
The given lines form tangent and normal to above ellipse.
Is there any other nice way to eliminate $m$ w/o using graphing ?
I have tried eliminating $m$ by solving the intersection point, but it's looking very messy. Thanks!

Solving the system you get
$x =  - \frac{m}{{\sqrt {{m^2} + 2} }},y = \frac{2}{{\sqrt {{m^2} + 2} }}$
Now squaring and adding you get
${x^2} + {y^2} = \frac{{{m^2}}}{{{m^2} + 2}} + \frac{4}{{{m^2} + 2}} = {\left( {\sqrt {\frac{{{m^2} + 4}}{{{m^2} + 2}}} } \right)^2}$
which is a circle with center (0,0) and radius $r = \left( {\sqrt {\frac{{{m^2} + 4}}{{{m^2} + 2}}} } \right)$
