Chapter 10: Q41. (page 677)

In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40. $16 x^{2} + 24 x y + 9 y^{2} - 130 x + 90 y = 0$

Short Answer

Expert verified

The parabola is rotated through an angle of $- 36 . 87^{\circ}$

Step by step solution

Step 1. Given Information

The given equation is $16 x^{2} + 24 x y + 9 y^{2} - 130 x + 90 y = 0$ .

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

Step 2. Finding acute angle

Here $A = 16, B = 24, C = 9$

localid="1646936800893" $c o t 2 θ = \frac{16 - 9}{24} = \frac{7}{24} 2 θ = - 73.74 θ = - 36.87$

Step 3. Finding expression for x,y

$x = x' \cos (- 36.87) - y' \sin (- 36.87) x = 0.79 x' + 0.6 y' a n d y = x' \sin (- 36.87) + y' \cos (- 36.87) y = - 0.6 x' + 0.79 y'$

Step 4. Substitute these values in the original equation

$16 x^{2} + 24 x y + 9 y^{2} - 130 x + 90 y = 0 16 {(0.79 x' + 0.6 y')}^{2} + 24 (0.79 x' + 0.6 y') (- 0.6 x' + 0.79 y') + 9 {(- 0.6 x' + 0.79 y')}^{2} - 130 (0.79 x' + 0.6 y') + 90 (- 0.6 x' + 0.79 y') = 0 16 (0.6 x'^{2} + 0.9 x' y' + 0.3 y'^{2}) + 24 (- 0.4 x'^{2} + 0.6 x' y' - 0.3 x' y' + 0.4 y'^{2}) + 9 (0.6 y'^{2} - 0.9 x' y' + 0.3 x'^{2}) - 102.7 x' - 78 y' - 54 x' + 71.1 y' = 0$

Step 5. Simplify above Equation

$9.6 x'^{2} + 14.4 x' y' + 4.8 y'^{2} - 9.6 x'^{2} + 7.2 x' y' + 9.6 y'^{2} + 5.4 y'^{2} - 8.1 x' y' + 2.7 x'^{2} - 102.7 x' - 78 y' - 54 x' + 71.1 y' = 0 2.7 x'^{2} + 13.5 x' y' + 19.8 y'^{2} - 156 x' - 6.9 y' = 0$

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.

In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40. $16 x^{2} + 24 x y + 9 y^{2} - 130 x + 90 y = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding expression for x,y

Step 4. Substitute these values in the original equation

Step 5. Simplify above Equation

Step 6. Graphing Equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.16x2+24xy+9y2-130x+90y=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding expression for x,y

Step 4. Substitute these values in the original equation

Step 5. Simplify above Equation

Step 6. Graphing Equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40. $16 x^{2} + 24 x y + 9 y^{2} - 130 x + 90 y = 0$