Rotation matrix to construct canonical form of a

Petrolovujhm

Petrolovujhm

Answered question

2022-04-01

Rotation matrix to construct canonical form of a conic
C:9x2+4xy+6y210=0.
I've found C is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've studied the characteristic polynomial 
p(t)=det9-t226-t  
The eigenvalue are t1=5,t2=10, with associated eigenvectors (1,2),(2,1). Thus I construct the rotation matrix R by putting in columns the normalized eigenvectors (taking care that det(R)=1):
R=1512-21  
Then (x,y)t=R(x,y)t, and after some computations I find the canonical form
12x2+45y2=1.

Answer & Explanation

lernarfnincln6g

lernarfnincln6g

Beginner2022-04-02Added 14 answers

Explanation:
If q(x,y)=9x2+4xy+6y210
and x=15(x+2y) and y=15(2x+y),
then q(x,y)=5x2+10y210. So, q(x,y)=012x2+y2=1.

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