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Summary of Steps for Rewriting the Equation
of an Ellipse from General to Standard Form

STEP 1:Collect like terms together
STEP 2:Move constant terms to the right hand side of the equal sign.
STEP 3:Factor out the coefficient of x2 from each x term.
Factor out the coefficient of y2 form each y term.
STEP 4:Complete the squares. Add numbers to both sides of the equation.
STEP 5:Simplify
STEP 6:Divide the entire equation by an appropriate number so that the right hand side becomes 1.
STEP 7:Simplify any fractions.
In the following set of questions you will practise writing an equation from general to standard form.

(x - h)2 + (y - k)2 = 1
a2
b2
green ellipse: 4x2 + 9y2 + 32x - 72y + 172 = 0
black ellipse:9x2 + y2 - 90x+ 189 = 0
blue ellipse: x2 + 16y2 + 4x + 96y + 132 = 0

Note that for each coloured ellipse, we were able to rewrite the equation from general to standard form.
In each case, once we had the standard form, we could create a red ellipse which was identical to the green, black or blue ellipse.
Thus we can conclude that the standard form and the general form of an ellipse are equivalent. They represent the same ellipse.

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