# Identify the conic with th e given equa­tion and give its equation in standard form 6x^2 - 4xy + 9y^2 - 20x - 10y - 5 = 0

Identify the conic with th e given equa­tion and give its equation in standard form $6{x}^{2}-4xy+9{y}^{2}-20x-10y-5=0$
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odgovoreh
Given $6{x}^{2}-4xy+9{y}^{2}-20x-10y-5=0$ All degenerate conic sections have equations of the form $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$.
$A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ where A,B,C,D,E and F are constants. If ${B}^{2}-4AC$ is less than zero, if a conic exists and A = C, it will be a circle If ${B}^{2}-4AC$ is less than zero, if a conic exists , it will be ellipse. If ${B}^{2}-4AC$ equals zero, if a conic exists, it will be a parabola. If ${B}^{2}-4AC$ is greater than zero, if a conic exists, it will be a hyperbola. Here in equation $6{x}^{2}-4xy+9{y}^{2}-20x-10y-5=0$, ${B}^{2}-4AC={4}^{2}-4\cdot 6\cdot 9$
$=16-216$
$=-200<0$ and $A\ne C$ Hence it is an ellipse