solve the system for x , y and z . Is there any smart trick to solve it?

Sarai Davenport

Sarai Davenport

Answered question

2022-06-09

solve the system for x, y and z. Is there any smart trick to solve it?
{ 2 a ( a x + b y ) + 2 c ( c x + d y ) + 2 z x = 0 2 b ( a x + b y ) + 2 d ( c x + d y ) + 2 z y = 0 x 2 + y 2 1 = 0
a , b , c , d R and ( x , y , z ) R 3 .

Answer & Explanation

popman14ee

popman14ee

Beginner2022-06-10Added 19 answers

Make the substitutions
a = R 1 cos ( ϕ 1 ) , b = R 1 sin ( ϕ 1 ) c = R 2 cos ( ϕ 2 ) , d = R 2 sin ( ϕ 2 ) x = cos ( θ ) , y = sin ( θ )
Eliminate z to get an equation in θ
sin ( 3 ( θ ϕ 2 ) ) R 2 3 + sin ( θ ϕ 2 ) R 2 3 + sin ( 3 ( θ ϕ 1 ) ) R 1 3 + sin ( θ ϕ 1 ) R 1 3 = 0

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