Find the equations of the lines tangent to the circle $x^2+y^2=r^2$ that pass through the point $(a,0)$.
My book explains that the equation of this line is $y=m(x-a)$ and then we make the substitution:
$$x^2+[m(x-a)]^2=r^2\\(1-m^2)x^2-2am^2x+a^2m^2-r^2=0$$
And then it says for that this equation of second degree have only one root, it's discriminant must be zero. And then, this gives us:
$$ \overbrace{4a^2m^4}^{b^2}-4\overbrace{(1+m^2)}^{a}\overbrace{(a^2m^2-r^2)}^{c}=0$$
I don't understand why finding the tangent lines ammount to computing a discriminant and solving for $m$. I know that making that substitution on $x^2+y^2=r^2$ is something that enables us to find the intersections of those curves, but I'm really lost at knowing what is the role the determinant plays here.