Question

9x^2-24xy+16y^2-20x-15y-50=0
Use axis rotation formulas for x and y to transform the quadratic equation to an equation in (u,v) coordinates with no cross product term. Identify the vertex or vertices in (x,y) coordinates
thank you so so so much

Answers

tan(2θ) = B/(A-C) = 24/7
so sinθ = 0.6 and cosθ = 0.8
Now apply the rotation matrix and you have
(x,y) =
(cosθ -sinθ) (u)
(sinθ cosθ) (v)

A' = Acos^2θ + Bsinθcosθ + Csin^2θ = .64A+.48B+.36C = .64*9-.48*24+.36*16 = 0
B' = (C-A)sin2θ + Bcos2θ = .96(C-A)+.28B = .96*7-.28*24 = 0
C' = Asin^2θ - Bsinθcosθ + Ccos^2θ = .36A-.48B+.64C = .36*9+.48*24+.64*16 = 25
D' = Dcosθ + Esinθ = .8D+.6E = -.8*20-.6*15 = 25
E' = -Dsinθ + Ecosθ = -.6D+.8E = .6*20-.8*15 = 0
F' = F = -50
and you end up with
25v^2+25u - 50 = 0
u = 2-v^2

see the graph at
https://www.wolframalpha.com/input/?i=9x%5E2-24xy%2B16y%5E2-20x-15y-50%3D0

Hmmm. I seem to have forgotten a - sign somewhere. The graph shows u = v^2-2 rotated clockwise, rather than the usual counter-clockwise. Maybe you can spot the mistake.

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