If the lines joining the origin to the intersection of the line $y=mx+2$ and the curve $x^2+y^2=1$ ae at right angles, then

Joint equation of the lines joining the origin and the point of intersection of the line $y=mx+2$ and the
curve $x^2+y^2=1$ is

$\begin{array}{l}{x}^2+y^2={\left(\frac{y-mx}{2}\right)}^2\\ \Rightarrow x^2\left(4-m^2\right)+2mxy+3y^2=0\end{array}$

Since these lines are at right angles

$4-m^2+3=0\Rightarrow m^2=7$