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

Question

identify the graph of the equation.

Answer 1

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 + y  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

Answer 2

Continuous...

The rotation of axes formula give

Substituting these in to original equation

After simplifying we have

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

= 15.66.