If the pair of lines ax^2+2hxy+by^2+2gx+2fy+c=0 intersect on the y-axis then:

Question

If the pair of lines $ax^2+2hxy+by^2+2gx+2fy+c=0$ intersect on the $y$-axis then:

Accepted Answer

Setting $x=0$: $by^2+2fy+c=0$. For the pair of lines to intersect on the $y$-axis (i.e. this quadratic has a repeated/unique intersection point consistent with the pair structure), the discriminant condition combined with the general pair-of-lines determinant condition
$abc+2fgh-af^2-bg^2-ch^2=0$
together with $af^2=abc$ (from the $x=0$ intersection condition $f^2=bc$, i.e. $af^2-abc=0$) gives:
$$2fgh-bg^2-ch^2=0 \implies 2fgh=bg^2+ch^2$$

Solutions · 1