Question Statement Find the measure of the angle through which the axis must be rotated so that the product term x y xy x y
i. 2 x 2 + 6 x y + 10 y 2 − 11 = 0 2x^2 + 6xy + 10y^2 - 11 = 0 2 x 2 + 6 x y + 10 y 2 − 11 = 0
ii. x y + 4 x − 3 y − 10 = 0 xy + 4x - 3y - 10 = 0 x y + 4 x − 3 y − 10 = 0
iii. 5 x 2 − 6 x y + 5 y 2 − 8 = 0 5x^2 - 6xy + 5y^2 - 8 = 0 5 x 2 − 6 x y + 5 y 2 − 8 = 0
Background and Explanation To solve these problems, we need to perform a coordinate rotation to eliminate the x y xy x y x x x y y y θ \theta θ
x ′ = x cos θ − y sin θ x' = x \cos \theta - y \sin \theta x ′ = x cos θ − y sin θ y ′ = x sin θ + y cos θ y' = x \sin \theta + y \cos \theta y ′ = x sin θ + y cos θ After substituting these transformed coordinates into the original equation, we will solve for θ \theta θ x y xy x y x y xy x y
Solution i. 2 x 2 + 6 x y + 10 y 2 − 11 = 0 2x^2 + 6xy + 10y^2 - 11 = 0 2 x 2 + 6 x y + 10 y 2 − 11 = 0
Equation of Transformation :
x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ x' = x \cos \theta - y \sin \theta, \quad y' = x \sin \theta + y \cos \theta x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ
Substitute into the given equation :
2 ( x cos θ − y sin θ ) 2 + 6 ( x cos θ − y sin θ ) ( x sin θ + y cos θ ) + 10 ( x sin θ + y cos θ ) 2 − 11 = 0 2(x \cos \theta - y \sin \theta)^2 + 6(x \cos \theta - y \sin \theta)(x \sin \theta + y \cos \theta) + 10(x \sin \theta + y \cos \theta)^2 - 11 = 0 2 ( x cos θ − y sin θ ) 2 + 6 ( x cos θ − y sin θ ) ( x sin θ + y cos θ ) + 10 ( x sin θ + y cos θ ) 2 − 11 = 0
Expand the terms and simplify:
2 ( x 2 cos 2 θ − 2 x y sin θ cos θ + y 2 sin 2 θ ) + 6 ( x 2 cos θ sin θ + x y ( cos 2 θ − sin 2 θ ) − y 2 sin θ cos θ ) + 10 ( x 2 sin 2 θ + 2 x y sin θ cos θ + y 2 cos 2 θ ) − 11 = 0 2\left(x^2 \cos^2 \theta - 2xy \sin \theta \cos \theta + y^2 \sin^2 \theta\right) + 6\left(x^2 \cos \theta \sin \theta + xy (\cos^2 \theta - \sin^2 \theta) - y^2 \sin \theta \cos \theta\right) + 10\left(x^2 \sin^2 \theta + 2xy \sin \theta \cos \theta + y^2 \cos^2 \theta\right) - 11 = 0 2 ( x 2 cos 2 θ − 2 x y sin θ cos θ + y 2 sin 2 θ ) + 6 ( x 2 cos θ sin θ + x y ( cos 2 θ − sin 2 θ ) − y 2 sin θ cos θ ) + 10 ( x 2 sin 2 θ + 2 x y sin θ cos θ + y 2 cos 2 θ ) − 11 = 0
To eliminate the x y xy x y , set the coefficient of x y xy x y
− 4 sin θ cos θ + 6 ( cos 2 θ − sin 2 θ ) + 20 sin θ cos θ = 0 -4 \sin \theta \cos \theta + 6 (\cos^2 \theta - \sin^2 \theta) + 20 \sin \theta \cos \theta = 0 − 4 sin θ cos θ + 6 ( cos 2 θ − sin 2 θ ) + 20 sin θ cos θ = 0 Simplify:
6 cos 2 θ + 16 sin θ cos θ = 0 6 \cos 2\theta + 16 \sin \theta \cos \theta = 0 6 cos 2 θ + 16 sin θ cos θ = 0 6 cos 2 θ + 8 sin 2 θ = 0 6 \cos 2\theta + 8 \sin 2\theta = 0 6 cos 2 θ + 8 sin 2 θ = 0 6 cos 2 θ = − 8 sin 2 θ 6 \cos 2\theta = -8 \sin 2\theta 6 cos 2 θ = − 8 sin 2 θ tan 2 θ = − 3 4 \tan 2\theta = -\frac{3}{4} tan 2 θ = − 4 3
Solve for θ \theta θ :
3 tan 2 θ − 8 tan θ − 3 = 0 3 \tan^2 \theta - 8 \tan \theta - 3 = 0 3 tan 2 θ − 8 tan θ − 3 = 0 Factor the equation:
( tan θ − 3 ) ( 3 tan θ + 1 ) = 0 (\tan \theta - 3)(3 \tan \theta + 1) = 0 ( tan θ − 3 ) ( 3 tan θ + 1 ) = 0 Solving for tan θ \tan \theta tan θ
tan θ = 3 ⇒ sin θ = 3 10 , cos θ = 1 10 \tan \theta = 3 \quad \Rightarrow \quad \sin \theta = \frac{3}{\sqrt{10}}, \quad \cos \theta = \frac{1}{\sqrt{10}} tan θ = 3 ⇒ sin θ = 10 3 , cos θ = 10 1
Substitute θ = 4 5 ∘ \theta = 45^\circ θ = 4 5 ∘ into the equation to find the transformed equation:
11 x 2 + y 2 − 11 = 0 11x^2 + y^2 - 11 = 0 11 x 2 + y 2 − 11 = 0 ii. x y + 4 x − 3 y − 10 = 0 xy + 4x - 3y - 10 = 0 x y + 4 x − 3 y − 10 = 0
Equation of Transformation :
x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ x' = x \cos \theta - y \sin \theta, \quad y' = x \sin \theta + y \cos \theta x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ
Substitute into the given equation :
( x cos θ − y sin θ ) ( x sin θ + y cos θ ) + 4 ( x cos θ − y sin θ ) − 3 ( x sin θ + y cos θ ) − 10 = 0 (x \cos \theta - y \sin \theta)(x \sin \theta + y \cos \theta) + 4(x \cos \theta - y \sin \theta) - 3(x \sin \theta + y \cos \theta) - 10 = 0 ( x cos θ − y sin θ ) ( x sin θ + y cos θ ) + 4 ( x cos θ − y sin θ ) − 3 ( x sin θ + y cos θ ) − 10 = 0
Eliminate the x y xy x y :
Set the coefficient of x y xy x y
cos 2 θ − sin 2 θ = 0 \cos^2 \theta - \sin^2 \theta = 0 cos 2 θ − sin 2 θ = 0 tan θ = 1 \tan \theta = 1 tan θ = 1 θ = 4 5 ∘ \theta = 45^\circ θ = 4 5 ∘
Substitute θ = 4 5 ∘ \theta = 45^\circ θ = 4 5 ∘ into the equation to find the transformed equation:
x 2 − y 2 + 2 x − 7 2 y − 20 = 0 x^2 - y^2 + \sqrt{2} x - 7\sqrt{2} y - 20 = 0 x 2 − y 2 + 2 x − 7 2 y − 20 = 0 iii. 5 x 2 − 6 x y + 5 y 2 − 8 = 0 5x^2 - 6xy + 5y^2 - 8 = 0 5 x 2 − 6 x y + 5 y 2 − 8 = 0
Equation of Transformation :
x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ x' = x \cos \theta - y \sin \theta, \quad y' = x \sin \theta + y \cos \theta x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ
Substitute into the given equation :
5 ( x cos θ − y sin θ ) 2 − 6 ( x cos θ − y sin θ ) ( x sin θ + y cos θ ) + 5 ( x sin θ + y cos θ ) 2 − 8 = 0 5(x \cos \theta - y \sin \theta)^2 - 6(x \cos \theta - y \sin \theta)(x \sin \theta + y \cos \theta) + 5(x \sin \theta + y \cos \theta)^2 - 8 = 0 5 ( x cos θ − y sin θ ) 2 − 6 ( x cos θ − y sin θ ) ( x sin θ + y cos θ ) + 5 ( x sin θ + y cos θ ) 2 − 8 = 0
Eliminate the x y xy x y :
Set the coefficient of x y xy x y
− 10 sin θ cos θ + 6 ( cos 2 θ − sin 2 θ ) + 10 sin θ cos θ = 0 -10 \sin \theta \cos \theta + 6 (\cos^2 \theta - \sin^2 \theta) + 10 \sin \theta \cos \theta = 0 − 10 sin θ cos θ + 6 ( cos 2 θ − sin 2 θ ) + 10 sin θ cos θ = 0 cos 2 θ − sin 2 θ = 0 ⇒ tan 2 θ = 1 ⇒ tan θ = 1 \cos^2 \theta - \sin^2 \theta = 0 \quad \Rightarrow \quad \tan^2 \theta = 1 \quad \Rightarrow \quad \tan \theta = 1 cos 2 θ − sin 2 θ = 0 ⇒ tan 2 θ = 1 ⇒ tan θ = 1 θ = 4 5 ∘ \theta = 45^\circ θ = 4 5 ∘
Substitute θ = 4 5 ∘ \theta = 45^\circ θ = 4 5 ∘ into the equation to find the transformed equation:
x 2 + 4 y 2 − 4 = 0 x^2 + 4y^2 - 4 = 0 x 2 + 4 y 2 − 4 = 0 x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ x' = x \cos \theta - y \sin \theta, \quad y' = x \sin \theta + y \cos \theta x ′ = x cos θ − y sin θ , y ′ = x sin θ + y cos θ
Eliminating the x y xy x y : Set the coefficient of the x y xy x y θ \theta θ
Summary of Steps
Define the coordinate transformations using the angle θ \theta θ Substitute the transformed coordinates into the given equation.Expand the equation and simplify.Set the coefficient of the x y xy x y to eliminate it.Solve for θ \theta θ and substitute it back into the equation to find the transformed equation. 
