lecture 36-draft.pdf - 1 AM 1411a Fall 2014 Lectures 36 December 1 Quadratic forms(Sec 7.3 Conic sections Conics in standard position A general

# lecture 36-draft.pdf - 1 AM 1411a Fall 2014 Lectures 36...

• No School
• AA 1
• 8

This preview shows page 1 - 8 out of 8 pages.

1 AM 1411a, Fall 2014 Lectures 36, December 1 Quadratic forms (Sec. 7.3) Conic sections

Subscribe to view the full document.

Conics in standard position
A general quadratic equation in R 2 : ax 2 1 + 2 bx 1 x 2 + cx 2 2 + dx 1 + ex 2 + f = 0 , where a; b; c; d; e; f are constants. It can be rewritten as h x 1 x 2 i " a b b c # " x 1 x 2 # + h d e i " x 1 x 2 # + f = 0 , or x T A x + v T x + f = 0 : Note: a matrix A is symmetric. Ex 1. Express 2 x 2 1 +3 x 1 x 2 ° 4 x 2 2 + x 1 +5 = 0 in matrix notation. h x 1 x 2 i " 2 3 2 3 2 ° 4 # " x 1 x 2 # + h 1 0 i " x 1 x 2 # +5 = 0 :

Subscribe to view the full document.

A conic in non-standard position To eliminate a cross product term (proportional to x 1 x 2 ) in x T A x + v T x + f = 0 ; use diagonalization of the matrix A ; to eliminate linear terms °shift a center of the coordinate system. Ex 2. Let 3 x 2 1 +2 x 1 x 2 +3 x 2 2 ° 8 = 0 : Identify the conic by eliminating the cross product term. h x 1 x 2 i " 3 1 1 1 # " x 1 x 2 # = 8
Diagonalize A = " 3 1 1 1 # : 1) det ( °I ° A ) = det " ° ° 3 ° 1 ° 1 ° ° 3 # = 0 ° 2 ° 6 ° + 8 = 0 ° 1 = 2 ; ° 2 = 4 2) (a) ° 1 = 2 ) (2 ± I ° A ) x = 0 " ° 1 ° 1 0 ° 1 ° 1 0 # RREF °°°°! " 1 1 0 0 0 0 # Let x 2 = t , then x 1 = ° t: x = " ° t t # = ° t " 1 ° 1 # ; p 1 = " 1 ° 1 # (b) ° 2 = 4 ) (4 I ° A ) x = 0 )

Subscribe to view the full document.

" 1 ° 1 0 ° 1 1 0 # RREF °°°°! " 1 ° 1 0 0 0 0 # Let x 2 = t , then x 1 = t: x = " t t # = t " 1 1 # ; p 2 = " 1 1 # p 1 ± p 2 = 0 ) orthogonal. Normalize them: q 1 = p 1 k p 1 k = 1 p 2 " 1 ° 1 # ; q 2 = p 2 k p 2 k = 1 p 2 " 1 1 # Q = 2 4 1 p 2 1 p 2 ° 1 p 2 1 p 2 3 5 is a transformation matrix that rep- resent rotation about the origin on 45

Subscribe to view the full document.

• Fall '19
• #, Conic section, 0

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes