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9x^2 + 5xy + y^2 = 3. What is the angle of rotation for the equation to the nearest degree?

0 votes
identify the graph of the equation.
asked Mar 5, 2014 in ALGEBRA 2 by rockstar Apprentice

2 Answers

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Rotating Quadratic equations

To eliminate the xy  term in the general quadratic equation.

Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 where B not equals to 0.

A = 9, B = 5, C = 1, D  = 0, E  = 0, F  = -3.

Coordinates rotation formulas

If a rectangular coordinate system is rotated through an angle Ѳ to from an x ' y ' cordinate system.

Where (x, y ) and (x ', y ') are related by

x  = x ' CosѲ - y ' SinѲ

y = x ' SinѲ + y ' CosѲ

And x ' = x cosѲ + sinѲ

y' = -x SinѲ + y CosѲ

Rotate the coordinate axes through an angle theta that satisifies Cot(2Ѳ ) = (A - C )/B

 

From half angle identities

answered Apr 24, 2014 by david Expert
edited Apr 24, 2014 by david
0 votes

Continuous...

The rotation of axes formula give

Substituting these in to original equation

After simplifying we have

image

image

This is graph of an ellipse that has been rotated through angle image = arc sin(0.27)

image = 15.66.

answered Apr 24, 2014 by david Expert
edited Apr 24, 2014 by david

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