2. a) Let D = [(x : y : z) | x2 - 2xy+ > =0). i) Classify as one of the 5 types of conics we studied. ii) Find a symmetric matrix A such that @ = 4. iii) Find the tangent Up for all points PED. b) Leto = [(x : y : z) |x'-2xz+y' =0), @, = ((x : y : z) |xy = z'). Let yy be a collineation with ww(1 : 0 : 0) = (0 :0 : 1), ww(0 : 1 : 0) = (1 : 1 : 1), Ww(1 : 1 : 1) = (2 :0 : 1). i) Find a symmetric matrix A such that = 4. ii) Find a matrix M such that yy(Po) = . iii) Find n /. and classify @ according to its intersection with &, (i.e. is it an ellipse or parabola or hyperbola). Solution a) i) It's the "fat" line: (1 : -1 : 0). 0 ii) A = 0 0 0 0 iii) P is either (1 : 1 : a) for some a or (0 : 0 : 1). In either case the tangent is P. 1 0 b) i) A = 0 1 0 -10 0 ii) The point (0 : 1 : 1) is the intersection of the tangent lines to @ at (0 : 0 : 1) and (1 : 1 : 1) ONO and therefore is equal to wy(0 : 0 : 1). Therefore we can assume M = 0 # v and H 0 H 0 H 18- 18 We can choose # = 2, # = -v, A+ # + v = 1. We obtain 10 2 0 M = 0 2 -2 2 iii) Substituting z = 0 produces x' + y' = 0, no solution in P. Therefore it's an ellipse