For what values of m and n does x^2+6xy+my^2+3x+ny-4=0…
For what values of $m$ and $n$ does $x^2+6xy+my^2+3x+ny-4=0$ represent: $1^\circ$ a central conic; $2^\circ$ a parabola; $3^\circ$ a pair of parallel lines?
Explanation of Method
Conditions on the parameters m and n for a central conic, a parabola, or a pair of parallel lines.
Step 1: Invariants
\[ \delta=m-9,\qquad \Delta=\frac{-25m-n^2+18n+144}{4}=\frac{225-25m-(n-9)^2}{4}. \]
Step 2: $1^\circ$ Central conic
A unique centre requires $\delta\ne0$: $m\ne9\quad(n\ \text{arbitrary})$.
More precisely:
- For $m\gt 9$, $\Delta\lt 0$ always, so the curve is a real ellipse.
- For $m\lt 9$ it is a hyperbola, or a pair of intersecting lines when $\Delta=0$.
Step 3: $2^\circ$ Parabola
With $m=9$: $\Delta=-\tfrac{(n-9)^2}{4}$.
\[m=9,\qquad n\ne9\]
Step 4: $3^\circ$ Parallel lines
With $m=9$ and $n=9$, set $t=x+3y$: $t^2+3t-4=(t+4)(t-1)=0$.
\[(x+3y+4)(x+3y-1)=0,\qquad d=\frac{5}{\sqrt{10}}=\frac{\sqrt{10}}2\]
Step 5: Intersecting lines
$\Delta=0$ iff $m=9-\tfrac{(n-9)^2}{25}$. For $n\ne9$ this gives $m\lt 9$, i.e. $\delta\lt 0$: a pair of intersecting lines. For example, $n=4$, $m=8$:
\[ x^2+6xy+8y^2+3x+4y-4=(x+2y-1)(x+4y+4)=0 . \]
Result
\[\boxed{\,m\ne9\quad(n\ \text{arbitrary})\,}\]
\[\boxed{\,m=9,\qquad n\ne9\,}\]
\[\boxed{\,(x+3y+4)(x+3y-1)=0,\qquad d=\frac{5}{\sqrt{10}}=\frac{\sqrt{10}}2\,}\]
Remark
Correction: the source's final remark (“$\Delta=0$ forces $m=9$, so intersecting lines cannot occur”) is false. $\Delta=0$ defines the whole curve $m=9-\tfrac{(n-9)^2}{25}$ in the $(m,n)$-plane, and for $n\ne9$ it gives intersecting lines.