Answer to Question #125312 in Analytic Geometry for Samuel kassapa

Question

A conic session is given by the equation 57x^2+14√3xy+43y^2-576=0

Answer 1

57x²+14(3)^1/2xy+43 y²-576=0

We give a quadratic form to the main axes, that is, to the canonical form. The matrix of this quadratic form:

$\begin{vmatrix} 57 & 14√3/2 \\ 14√3/2 & 43 \end{vmatrix}$

We find the eigenvalues ​​and eigenvectors of this matrix

The characteristic equation:

$\begin{vmatrix} 57-l & 7√3 \\ 7√3 & 43-l \end{vmatrix}$ =(57-l)*(43-l)-147=l²-100l+2304=0

l₁=36 l₂=64

replace the original equation 36x₁²+64y₁² -576=0

x₁²/16+y₁²/9=1

canonical ellipse equation