Discriminant is Invariant for Isometry of Conic Section

Theorem

Let a conic section $\KK$ be expressed as a quadratic form in $2$ variables:

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

The discriminant:

$b^2 - 4 a c$

is an invariant under translations and rotations of the coordinate axes.

Proof

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Sources

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