0

Is it true that if $\Delta \neq 0$ and $h^2 < ab$ we can have either empty points on the curve $ax^2 + by^2 + 2hxy + 2gx + 2fy + c= 0$ or the ellipse ? I know that in case of $\Delta = 0$ $h^2 < ab$ would represent a single point but how do we derive these conditions for single and empty points ? $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 $

ProblemDestroyer
  • 967
  • I believe, you need to add information what are $a$, $b$, $c$, $f$, $g$, $h$ and how they are related to ellipse or curve? – Ivan Kaznacheyeu Apr 18 '22 at 07:12
  • Added :)Ivan Kaznacheyeu – ProblemDestroyer Apr 18 '22 at 11:37
  • The claim is true. See table in the end of following article: Second-order curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Second-order_curve&oldid=23972 To prove it, one can prove fact, that curve is limited at condition $h^2 < ab$. The only types of limited curves are ellipse and point (or empty set). – Ivan Kaznacheyeu Apr 18 '22 at 12:25

0 Answers0