Q.Show that the lines $x^2 - 4xy + y^2 = 0$ and $x + y = 10$ contain the sides of an equilateral triangle. Find the area of the triangle.

The combined equation of any two lines (not necessarily through the origin) always has the fully general second-degree shape $ax^2+2hxy+by^2+2gx+2fy+c=0$, but the converse fails -- equations like $x^2+y^2=25$ have the same shape yet describe a circle, not two lines. A general second-degree equation genuinely represents a pair of lines only when both $abc+2fgh-af^2-bg^2-ch^2=0$ (equivalently, the $3\times3$ determinant with $a,b,c$ on the diagonal and $h,g,f$ off it vanishes) and $h^2-ab\geq0$ hold together; in practice the safest way to confirm this and find the two lines is to factor the quadratic part and match the remaining linear/constant terms by comparison. Once confirmed, the pair is parallel to the lines of its homogeneous part $ax^2+2hxy+by^2=0$, so the same angle formula, perpendicularity condition $a+b=0$, and parallel condition $h^2-ab=0$ carry over unchanged, and (when $h^2-ab>0$) the two lines meet at $\left(\df …