General form of conic sections:
A x² + B x y + C y² + D x + E y + F = 0
Now identifying conic sections by the discriminant.
1.
If A = C then the graph is a circle
2.
If B² - 4 A C < 0 then the graph is an ellipse.
3.
If B² - 4 A C > 0 then the graph is a hyperbola.
4.
If B² - 4 A C = 0 then the graph is a parabola.
In this case;
2 x² - 4 x y + 5 y² = 6
Subtract 6 to both sides:
2 x² - 4 x y + 5 y² - 6 = 0
In equation:
A x² + B x y + C y² + D x + E y + F = 0
coefficients are:
A = 2 , B = - 4 , C = 5 , D = 0 , E = 0 , F = 6
B² - 4 A C = ( - 4 )² - 4 ∙ 2 ∙ 5 = 16 - 40 = - 24 < 0
The graph is an ellipse.
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1 answer
1 answer