I have the general equation \(\displaystyle{A}{x}^{{2}}+{B}{x}{y}+{C}{y}^{{2}}+{D}{x}+{E}{y}+{F}={0}\) and I want

ab0utfallingm1z2

ab0utfallingm1z2

Answered question

2022-04-01

I have the general equation
Ax2+Bxy+Cy2+Dx+Ey+F=0
and I want to determine what conic section it represents according to the value of its coefficients or relations between them, is there a theorem where that is explained?

Answer & Explanation

Laylah Hebert

Laylah Hebert

Beginner2022-04-02Added 15 answers

This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as
Q=(x,y)(AB/2B/2C)(x,y)T+(D E)(x,y)T+F=0.
the first matrix X is called the matrix of the quadratic form.
Q is a hyperbola if and only if det A<0,
Q is a parabola if and only if det A=0, and
Q is an ellipse if and only if det A>0.

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