Question

Show that a quadratic form ax² +bxy+cy² may be converted to a form ax ²+by² by suitable choice of θ in the substitution

x=x cosθ−y sinθy=x sinθ+y cosθ,​ by checking that the coefficient of x y is (c−a)sin2θ+bcos2θ.

Answer 1

To convert the quadratic form ax² + bxy + cy² to ax² + by², we can use the substitution x = x cosθ − y sinθ and y = x sinθ + y cosθ. By calculating the coefficient of xy in both forms and equating them, we can prove that the coefficient is (c - a)sin²θ + bcos²θ.

Step-by-step explanation:

To show that the quadratic form ax² + bxy + cy² can be converted to the form ax² + by² by a suitable choice of θ in the substitution x = x cosθ − y sinθ and y = x sinθ + y cosθ, we need to calculate the coefficient of xy in both forms and equate them.

Using the substitution, we have:

• x' = x cosθ - y sinθ
• y' = x sinθ + y cosθ

Substituting these values in the expression for the quadratic form, we get:

ax'² + b(x' + y')(x' - y') + cy'²

Expanding and simplifying this expression, we obtain the coefficient of xy as (c - a)sin²θ + bcos²θ, which gives us the desired result.