Past exam of the short options course of the University of Oxford / 2014 / 3 / d

For the explicit basis in part (c), $q_0+t{q}_1$ has coefficients $(1+t,1-2t,t-2)$. ThusAt these values the conics factor, up to nonzero constants, asGeometrically these are exactly the three pairings of the four base points into two pairs, with a projective line joining each pair. No double line can contain four points in general linear position.

The numerical parameter values depend on the selected basis, but the three distinct singular members do not. Another pair of independent nonsingular basis conics gives an invertible projective change of pencil parameter. Its parameter infinity is nonsingular because the second basis form is nonsingular. Therefore all three singular members again occur at three distinct finite values of $t$. This proves the claimed count for any such basis.