If the lines joining the origin to the points of intersection of the line $y = mx + 1$ and the circle $x^2 + y^2 = 1$ are perpendicular to each other, then the value of $m^2$ is:
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Homogenization reduces the perpendicularity condition of the intersecting lines to simply setting the sum of the $x^2$ and $y^2$ coefficients to zero: $(1-m^2) + 0 = 0 \implies m^2 = 1$.