Chapter 10

Section 6

Transformations Of Conics

by: Wells S.  in Ms. Blackwell's Honors ICM Class

Identify the graph of the equation. Then rotate the graph by the given angle and give the new equation.

9x² + 4y² =36 ,  rotate 30^o

First find the discriminate to determine what the graph is:

0² + 4(9)(4)                                           : because b=0, then it is an ellipse

Determine the appropriate values for x and y:

SQUARE ROOT = ?

x = x cos30^o+ y sin30^o      or      x = ?3x/2 + 1y/2

   y = -x sin30^o + y cos30^o      or      y= -1x/2 + ?3y/2 

Plug in the new values for x and y into the formula:

SQUARE ROOT = ?

9( ?3x/2 + 1y/2)² + 4( -1x/2 + ?3y/2)² = 36

Expand the Binomials:

careful with your algebra

SQUARE ROOT = ?

9( 3x²/4 + ?3xy/4 + y²/4) + 4( x²/4 - 2?3xy/4 + 3y²/4) = 36

Factor out the 4's:

SQUARE ROOT = ?

9/4( 3x² + ?3xy + y²) + 4/4( x²- 2?3xy + 3y²) = 36

Multiply both sides by 4:

SQUARE ROOT = ?

9( 3x² + ?3xy + y²) + 4( x²- 2?3xy + 3y²) = 144

Simplify:

SQUARE ROOT = ?

27x² + 9?3xy + 9y² + 4x² - 8?3xy + 12y² = 144

THEN:

31x² + ?3xy + 21y² = 144

The equation of the ellipse after the 30^o rotation is:

SQUARE ROOT = ?

31x² + ?3xy + 21y² = 144

"Life is a series of problems. Do you want to moan about them or solve them?"
M. Scott Peck  

http://www.usd259.com/charactered/quotes.htm

M. Scott Peck is making a good point about life, you can either complain about it or live it.  His quote is in concurrence with the character trait of perseverance.

A link to a related site that can help you with further instruction of graphing ellipses. http://johnbanks.maths.latrobe.edu.au/Games/Ellipse/

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