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

Answer 1

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.