Chapter 10: Q. 10 (page 700)

Given the equation $41 x^{2} - 24 x y + 34 y^{2} - 25 = 0$ rotate the axes so that there is no $x y$ -term. Analyze and graph the new equation.

Short Answer

Expert verified

The equation represent an ellipse with a rotation of $0.99 °$

Step by step solution

Step 1. Given Information

The given equation is $41 x^{2} - 24 x y + 34 y^{2} - 25 = 0$

We need to determine the appropriate rotation so that the new equation does not contain $x y$ term.

Step 1. Finding acute angle

Here $A = 41, B = - 24, C = 34$

$\cot (2 θ) = \frac{41 - 34}{- 24} = - \frac{7}{24}$

where role="math" localid="1650435128455" $0 ° < 2 θ < 180 °$

$2 θ = c o t^{- 1} (- \frac{7}{24}) 2 θ = 1.85 θ = 0.925$

Step 3. Finding x and y

$x = x' \cos θ - y' \sin θ x = x' \cos (0.925) - y' \sin (0.925) x = 0.9 x' - 0.9 y'$

and

$y = x' \sin (0.925) + y' \cos (0.925) y = 0.9 x' + 0.9 y'$

Step 4. Substituting these values into the original equation

$41 x^{2} - 24 x y + 34 y^{2} - 25 = 0 41 {(0.9 x' - 0.9 y')}^{2} - 24 (0.9 x' - 0.9 y') (0.9 x' + 0.9 y') + 34 {(0.9 x' + 0.9 y')}^{2} - 25 = 0 41 (0.81 x'^{2} - 1.62 x' y' + 0.81 y'^{2}) - 24 (0.81 x'^{2} - 0.81 y'^{2}) + 34 (0.81 x'^{2} + 1.62 x' y' + 0.81 y'^{2}) - 25 = 0 33.21 x'^{2} - 66.42 x' y' + 33.21 y'^{2} - 19.44 x'^{2} + 19.44 y'^{2} + 27.54 x'^{2} + 55.08 x' y' + 27.54 y'^{2} - 25 = 0 41.31 x'^{2} - 11.34 x' y' + 80.19 y'^{2} - 25 = 0$

Step 6. Graphing equation

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Given the equation $41 x^{2} - 24 x y + 34 y^{2} - 25 = 0$ rotate the axes so that there is no $x y$ -term. Analyze and graph the new equation.

Short Answer

Step by step solution

Step 1. Given Information

Step 1. Finding acute angle

Step 3. Finding x and y

Step 4. Substituting these values into the original equation

Step 6. Graphing equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Mechanics Maths

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Given the equation 41x2-24xy+34y2-25=0 rotate the axes so that there is no xy-term. Analyze and graph the new equation.

Short Answer

Step by step solution

Step 1. Given Information

Step 1. Finding acute angle

Step 3. Finding x and y

Step 4. Substituting these values into the original equation

Step 6. Graphing equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Mechanics Maths

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Given the equation $41 x^{2} - 24 x y + 34 y^{2} - 25 = 0$ rotate the axes so that there is no $x y$ -term. Analyze and graph the new equation.