Answer to Question #125312 in Analytic Geometry for Samuel kassapa

Question #125312
A conic session is given by the equation 57x^2+14√3xy+43y^2-576=0
1
Expert's answer
2020-07-06T20:12:00-0400

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}\n 57 & 14\u221a3\/2 \\\\\n 14\u221a3\/2 & 43\n\\end{vmatrix}"


We find the eigenvalues ​​and eigenvectors of this matrix

The characteristic equation:

"\\begin{vmatrix}\n 57-l & 7\u221a3 \\\\\n 7\u221a3 & 43-l\n\\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

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