Unformatted text preview: ( iv ) ±ind all the ²xed points of φ . 4. ±or each of the following conics, ²nd an associated symmetric matrix and show that the conic is nonsingular. ±ind all points where ℓ ∞ meets the conic and say whether the conic is an ellipse, a hyperbola or a parabola. ( i ) x 2 + y 2z 2 + 2 xy2 yz + 4 zx = 0 ( ii ) x 2y 2 + 2 xy2 yz + 4 zx = 0 ( iii ) x 2 + 4 y 22 xy + 2 z 2 + 2 yz6 zx = 0 5. ±or all real λ, let C ( λ ) denote the conic x 2 + 2 λxy + y 217 10 xz17 10 yz2 λz 2 = 0 . Show that ( i ) C ( λ ) passes through each of the points (4 : 1 : 2) and (1 : 4 : 2), for all λ . ( ii ) C ( λ ) is a hyperbola if  λ  > 1. ( iii ) C ( λ ) is an ellipse if  λ  < 1. ( iv ) C ( λ ) is a parabola if λ =1. ( v ) C ( λ ) is a singular conic if λ = 1. ( vi ) C ( λ ) is a circle if λ = 0....
View
Full Document
 Three '11
 ?
 Geometry, Topology, Conic section, Real projective plane, associated symmetric matrix

Click to edit the document details