by
Samuel Obara
Proof that the equation 
Is ellipse
The general equation
of a conic section is 
It should be noted that the
ellipse is not symetrical neither in the x-axis nor the y-axis
therefore we need a transformation so that it can be symetrical
along the x and y axis with center (0,0).
To transform the equation,
we need to use the following equations
which means the values
of X and Y will be
therefore in the equation
A = 1, B = 1 C = 1 by using the discriminant
then 
substitute x and y
in
it becomes
(x')^2 - 2x'y' + (y')^2
+ (x')^2 - (y')^2 + (x')^2 + 2x'y' + (y')^2 = 2
therefore
3(x')^2 + (y')^2 =
2 which means
{3(x')^2}/2 + {(y')^2}/2
= 1
(x')^2}/(2/3) + {(y')^2}/2
= 1
which
agrees with the general equation of ellipse:
note if we plot the
two equations, we obtain the following graph.
Which means the equation
is ellipse.
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