Question 249588
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The general equation of a conic section is:

{{{Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0}}}

The discriminant {{{B^2-4AC}}} is used to determine which 
conic section will result. 

1. If the discriminant is less than zero we have 
   (a). a circle if B = 0 and A = C, or 
   (b). an ellipse otherwise.
2. If the discriminant is equal to zero we have a parabola.
3. If the discriminant is greater than zero we have a hyperbola.


Your equation is:

{{{x^2/a^2+y^2/a^2-(2xy/a^2)Cos("45°")=Sin^2("45°")}}}

We put the term in {{{xy}}} in between the terms in {{{x^2)
and {{{y^2}}}, and get 0 on the right, so it will be in 
the same form as above:

{{{x^2/a^2-(2xy/a^2)Cos("45°")+y^2/a^2-Sin^2("45°")=0}}}

we have  

{{{A=1/a^2}}},

{{{B=-(2cos("45°"))/a^2=-2(sqrt(2)/2)/a^2 = -sqrt(2)/a^2}}}

{{{C = 1/a^2}}}

{{{D=0}}},  {{{E=0}}},  {{{F=-Sin^2("45°")}}}

So the discriminant = 

{{{B^2-4AC = (-sqrt(2)/a^2)^2-4(1/a^2)(1/a^2)=2/a^4-4/a^4=-2/a^4}}}  

Since the discriminant is negative, by the rules at the
top we can conclude the equation represents an ellipse.

Edwin</pre>