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Chapter 10: Q. 10 (page 700)

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

Expert verified

The equation represent an ellipse with a rotation of0.99掳

Step by step solution

01

Step 1. Given Information

The given equation is 41x2-24xy+34y2-25=0

We need to determine the appropriate rotation so that the new equation does not contain xyterm.

02

Step 1. Finding acute angle

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

cot2胃=41-34-24=-724

where role="math" localid="1650435128455" 0掳<2胃<180掳

2胃=cot-1-7242胃=1.85胃=0.925

03

Step 3. Finding x and y

x=x'cos胃-y'sin胃x=x'cos0.925-y'sin0.925x=0.9x'-0.9y'

and

y=x'sin0.925+y'cos0.925y=0.9x'+0.9y'

04

Step 4. Substituting these values into the original equation 

41x2-24xy+34y2-25=0410.9x'-0.9y'2-240.9x'-0.9y'0.9x'+0.9y'+340.9x'+0.9y'2-25=0410.81x'2-1.62x'y'+0.81y'2-240.81x'2-0.81y'2+340.81x'2+1.62x'y'+0.81y'2-25=033.21x'2-66.42x'y'+33.21y'2-19.44x'2+19.44y'2+27.54x'2+55.08x'y'+27.54y'2-25=041.31x'2-11.34x'y'+80.19y'2-25=0

05

Step 6. Graphing equation 

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