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Mathematics-Online problems:

Interactive Problem 45: Normal Form of a Quadric

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Given the following matrix in $ \mathbb{R}^2$ :

$\displaystyle Q: 3x_1^2+6x_2^2-4x_1x_2-8\sqrt{5}\,x_1-4\sqrt{5}\,x_2+49=0. $

Give the matrix representation of $ Q$ and find the Euclidian normal form.

Matrix representation: $ Q: x^{\rm {t}}Ax+2a^{\rm {t}}x+c=0$ ,         with

$ A=\left(\rule{0pt}{4ex}\right.$
$ \left.\rule{0pt}{4ex}\right)$ $ a=\sqrt{5}\left(\rule{0pt}{4ex}\right.$
$ \left.\rule{0pt}{4ex}\right)$ $ c=$

Euclidian normal form (give the coefficients in ascending order):

$ Q': $ $ z_1^2 + $ $ z_2^2 + 1 =0$ .

Mark of which geometric type $ Q$ is:

n/a empty set point line pair of lines
triangle ellipse hyperbola parabola prezel

Of what type is $ Q$ , if the Euclidian normal form is considered to be in $ \mathbb{R}^3$ ?

n/a ellipsoid pair of parallel planes
prism elliptic paraboloid two-sheeted hyperboloid
double cone empty set elliptic cylinder
cuboid prezeloid pair of intersecting planes


   

(Authors: App/Apprich/Höfert)
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  automatically generated: 8/11/2017