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Tangent Line Circle

Result

(b/sqrt(a^2 + b^2) | -a/sqrt(a^2 + b^2) -sin(t) | cos(t))

Example plots

Example plots Line

Example plots Circle

Equations

line | x(t) = b t y(t) = a (-t) - c/b circle | x(t) = a cos(t) y(t) = a sin(t)

line | a x + b y + c = 0 circle | x^2 + y^2 = a^2

line | r(θ) = -c/(a cos(θ) + b sin(θ)) circle | r(θ) = a

Characteristic polynomial

-(b λ)/sqrt(a^2 + b^2) - (a sin(t))/sqrt(a^2 + b^2) + (b cos(t))/sqrt(a^2 + b^2) + λ^2 - λ cos(t)

Eigenvalues

λ_1 = (sqrt(a^2 + b^2) cos(t) - sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2))

λ_2 = (sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2))

Eigenvectors

v_1 = (-(-sqrt(a^2 + b^2) cot(t) + b csc(t) - csc(t) sqrt((4 a sqrt(a^2 + b^2) - 2 b sqrt(a^2 + b^2) cot(t) + a^2 cos(t) cot(t) + b^2 cos(t) cot(t) + b^2 csc(t)) sin(t)))/(2 sqrt(a^2 + b^2)), 1)

v_2 = (-(-sqrt(a^2 + b^2) cot(t) + b csc(t) + csc(t) sqrt((4 a sqrt(a^2 + b^2) - 2 b sqrt(a^2 + b^2) cot(t) + a^2 cos(t) cot(t) + b^2 cos(t) cot(t) + b^2 csc(t)) sin(t)))/(2 sqrt(a^2 + b^2)), 1)

Diagonalization

M = S.J.S^(-1) where M = (b/sqrt(a^2 + b^2) | -a/sqrt(a^2 + b^2) -sin(t) | cos(t)) S = ((sqrt(a^2 + b^2) cot(t) + csc(t) sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) - b csc(t))/(2 sqrt(a^2 + b^2)) | cot(t)/2 - (csc(t) (sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b))/(2 sqrt(a^2 + b^2)) 1 | 1) J = ((sqrt(a^2 + b^2) cos(t) - sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2)) | 0 0 | (sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(a^2 + b^2))) S^(-1) = ((sqrt(a^2 + b^2) sin(t))/sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) | (-sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) + b)/(2 sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2)) -(sqrt(a^2 + b^2) sin(t))/sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) | (sqrt(a^2 + b^2) cos(t) + sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2) - b)/(2 sqrt(4 a sqrt(a^2 + b^2) sin(t) + (a^2 + b^2) cos^2(t) - 2 b sqrt(a^2 + b^2) cos(t) + b^2)))

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