# 1: Consider the following curves in R la) 1822-32 x y + 37 U2 100. l ) 2x2 + 6 x y + 2 y-100. 1c) x2 + 4 x y + 4 y2-10:0. Write them in normal form. Give the change of variables that does this....
# 1: Consider the following curves in R la) 1822-32 x y + 37 U2 100. l ) 2x2 + 6 x y + 2 y-100. 1c) x2 + 4 x y + 4 y2-10:0. Write them in normal form. Give the change of variables that does this. For example, in 1a) the orthonormal basis of eigenvectors are λί 5,V1 (2,1)'/V5 and λ2 = St ( 100. ) . That is, 45, ½ = (1,-2)t/V5.S ( 1/V 5-2/v/5 ) (V6, V(-2u)/V. Show that the equation is 5 U2 +45v2 So change to ( o here S = For 1b) use the same method of eigenvectors to render it in normal form. Hint: For the difficult curve in 1c) proceed as follows: The orthogonal basis of eigenvectors is so equation becomes 0 . U2 + 51/2-10x-0 or V2-10x = 0. Now express x in terms of U, V and complete the square to absorb the extra U term into V Are they ellipses, hyperbolas, parabolas? Hint: use wolframalpha.com but make sure that the eigenvectors are made to have length one so P is orthogonal.] 1 d) Can you make a rough plot of these curves in R2? If possible for the ellipses and hyperbolas indicate the minor and major axes llongest and shortest their directions, and places where the ellipse cuts the axes. If possible for the parabolas indicate the axis and the axis of symmetry and where they cross.
# 1: Consider the following curves in R la) 1822-32 x y + 37 U2 100. l ) 2x2 + 6 x y + 2 y-100. 1c) x2 + 4 x y + 4 y2-10:0. Write them in normal form. Give the change of variables that does this. For example, in 1a) the orthonormal basis of eigenvectors are λί 5,V1 (2,1)'/V5 and λ2 = St ( 100. ) . That is, 45, ½ = (1,-2)t/V5.S ( 1/V 5-2/v/5 ) (V6, V(-2u)/V. Show that the equation is 5 U2 +45v2 So change to ( o here S = For 1b) use the same method of eigenvectors to render it in normal form. Hint: For the difficult curve in 1c) proceed as follows: The orthogonal basis of eigenvectors is so equation becomes 0 . U2 + 51/2-10x-0 or V2-10x = 0. Now express x in terms of U, V and complete the square to absorb the extra U term into V Are they ellipses, hyperbolas, parabolas? Hint: use wolframalpha.com but make sure that the eigenvectors are made to have length one so P is orthogonal.] 1 d) Can you make a rough plot of these curves in R2? If possible for the ellipses and hyperbolas indicate the minor and major axes llongest and shortest their directions, and places where the ellipse cuts the axes. If possible for the parabolas indicate the axis and the axis of symmetry and where they cross.