Question 4.8.7: Rotation of a Conic Section Perform a rotation of axes to el...

The angle of rotation is given by

cot 2θ = \frac{a – c}{b} = \frac{5 – 5}{-6} = 0.

This implies that θ = π/4. So,

sin θ = \frac{1}{√2}   and    cos θ = \frac{1}{√2}.

By substituting

x = x´ cos θ  –  y´ sin θ = \frac{1}{√2}(x´ – y´)

and

y = x´ sin θ  +  y´ cos θ = \frac{1}{√2}(x´ +  y´)

into the original equation and simplifying, you obtain

Finally, by completing the square, you find the standard form of this equation to be

which is the equation of an ellipse, as shown in Figure 4.22.