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


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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