Mathematics · Ch 5 — Two Dimensional Analytical Geometry-II

Identifying the Conic from the General Equation

5.4.3

Identifying the Conic from the General Equation

Every conic — non-degenerate or degenerate — is some instance of $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ . Rather than always completing the square, the following coefficient patterns identify the type directly (for the axis-aligned case $B=0$ , the one used throughout this chapter's exercises):

#	Condition	Result
1	$A=C\ne0$ , $B=0$	Circle: $(x-h)^2+(y-k)^2=r^2$ with $h=-D/2A,\ k=-E/2A,\ r^2=h^2+k^2-F/A$
2	$B=0$ and exactly one of $A,C$ is $0$	Parabola
3	$A\ne C$ , same sign, $B=0$	Ellipse
4	$A\ne C$ , opposite signs, $B=0$	Hyperbola
5	$A=C$ , $B=D=E=0$ , $F=0$	A single point ( $x^2+y^2=0$ )
6	$A=C=F$ , $B=D=E=0$	An empty set ( $x^2+y^2+1=0$ , no real solution)
7	$A\ne0$ or $C\ne0$ , everything else zero	The coordinate axes
8	$A=-C$ , everything else zero	A pair of intersecting lines ( $x^2-y^2=0$ )

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