Quadratic Form in Two Variables represents Conic Section

Theorem

A quadratic form in $2$ variables, when put equal to a constant

$a x^2 + b x y + c y^2 = r$

represents a conic section.

Proof

It can be seen directly that:

$a x^2 + b x y + c y^2 = r$

is an instance of an equation for a conic section in Cartesian form:

$a x^2 + b x y + c y^2 + d x + e y + f = 0$

by setting $f \gets -r$ and equating $d$ and $e$ to zero.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): form
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): form