Ellipses Page 5

Page 5

Rewriting the Equation of an Ellipse from Standard to General Form

The standard equation of any ellipse can be rewritten into the following form:Ax² + By² + Cx + Dy + F = 0.

This general form can be obtained by expanding the standard equation of an ellipse.

Let's look at a few examples to see how this is done.

EXAMPLE 1: An ellipse centred at (3, 1) can be descibed by the standard equation (x - 3)²/ 9 + (y - 1)²/ 4 = 1.

Let us rewrite this equation into the form Ax² + By² + Cx + Dy + F = 0

Equation to expand (x - 3)²/ 9 + (y - 1)²/ 4 = 1

Multiply each term by the lowest common denominator, 36 36(x - 3)²/ 9 + 36(y - 1)²/ 4 = 36

Simplify the fractions 4(x - 3)² + 9(y - 1)² = 36

Expand the squared terms 4(x² - 6x + 9) + 9(y² - 2y + 1) = 36

Multiply 4x² - 24x + 36 + 9y² - 18y + 9 = 36

Write in the form Ax² + By² + Cx + Dy + F = 0 4x² + 9y² - 24x - 18y + 9 = 0

EXAMPLE 1:	An ellipse centred at (3, 1) can be descibed by the standard equation (x - 3)²/ 9 + (y - 1)²/ 4 = 1.
Let us rewrite this equation into the form Ax² + By² + Cx + Dy + F = 0
Equation to expand	(x - 3)²/ 9 + (y - 1)²/ 4 = 1
Multiply each term by the lowest common denominator, 36	36(x - 3)²/ 9 + 36(y - 1)²/ 4 = 36
Simplify the fractions	4(x - 3)² + 9(y - 1)² = 36
Expand the squared terms	4(x² - 6x + 9) + 9(y² - 2y + 1) = 36
Multiply	4x² - 24x + 36 + 9y² - 18y + 9 = 36
Write in the form Ax² + By² + Cx + Dy + F = 0	4x² + 9y² - 24x - 18y + 9 = 0

EXAMPLE 2: The standard equation x²/ 3 + y²/ 9 = 1 describes an ellipse centred at the origin.

Let us see what happens when we expand this into an equation of the form Ax² + By² + Cx + Dy + F = 0

Equation to expand x²/ 3 + y²/ 9 = 1

Multiply each term by the lowest common denominator, 9 9x²/ 3 + 9y²/ 9 = 9

Simplify the fractions 3x² + y² = 9

Write in the form Ax² + By² + Cx + Dy + F = 0 3x² + y² - 9 = 0

EXAMPLE 2:	The standard equation x²/ 3 + y²/ 9 = 1 describes an ellipse centred at the origin.
Let us see what happens when we expand this into an equation of the form Ax² + By² + Cx + Dy + F = 0
Equation to expand	x²/ 3 + y²/ 9 = 1
Multiply each term by the lowest common denominator, 9	9x²/ 3 + 9y²/ 9 = 9
Simplify the fractions	3x² + y² = 9
Write in the form Ax² + By² + Cx + Dy + F = 0	3x² + y² - 9 = 0

Notice that the general equation of this ellipse in Example 2 has no Cx or Dy terms. This is true for all ellipses centred at the origin.

The following are exercises for you to practice writing the standard equation of an ellipse as a general equation.