Given \(6x^2 - 4xy + 9y^2 - 20x - 10y - 5 = 0\)
All degenerate conic sections have equations of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).

\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) 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 \(6x^2 - 4xy + 9y^2 - 20x -10y - 5 = 0\), \(B^2 - 4 AC = 4^2 - 4 \cdot 6 \cdot 9\)

\(= 16 - 216\)

\(= - 200 < 0\) and \(A \neq C\) Hence it is an ellipse

\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) 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 \(6x^2 - 4xy + 9y^2 - 20x -10y - 5 = 0\), \(B^2 - 4 AC = 4^2 - 4 \cdot 6 \cdot 9\)

\(= 16 - 216\)

\(= - 200 < 0\) and \(A \neq C\) Hence it is an ellipse