Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² +2bxy+ay2² = = c with b 0 and c > 0. b (8). b a Let (x, y) = ax² + 2bry + ay2 and denote A = Note that: V(x,y) = (x v) 4 (4) = ( (4), 4 (*)) A where (,): R2 x R2 → R is the standard inner (dot) product on R². (a) Determine all eigenvalues of A, and their corresponding eigenspaces. (b) Construct an orthonormal basis u, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector () ER2 as a linear combination of u and v. (c) Use result from (b) to transform V(x, y) = ((*), 4 (1)) some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into (u, v) =au²+ Bv² for

If ƒ (§) = $c 4z²+z+5 dz, where C is the ellipse 12+ 2 = 1 z- 4 9 Find ƒ(i),ƒ(i),ƒ'(−1) and ƒ"(−1)

2. Rotate the coordinate axes to remove the xy-term. Then identify the type of conic and sketch its graph. (a) x² + 4xy−2y² −6=0 (b) x² +2√3xy+3y² +2√3x-2y=0

Exercise 11.6. Find two different ellipses passing through the four points (1, 1), (−1, 1), (−1, −1), and (1, -1).