Question

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

7x2 + 26xy + 7y2 - 24 = 0

Answer 1

This should help you answer this question: http://en.wikipedia.org/wiki/Rotation_of_axes

Answer 2

I've looked at that, but I'm not sure how to apply it to my question.

Answer 3

If you read through that web page you'll find it actually shows you the answer to your question.

Answer 4

Look at the section titled: Elimination of the xy term by the rotation formula

Answer 5

But I have tried using that formula and either I am not doing it correctly or I do not understand it. That is why I need help.

Answer 6

which part do you not understand?

Answer 7

I'm not quite sure... when I tried to plug in the A-C, I got 0, which does not help me to get an answer

Answer 8

A-C=0 is the correct approach

Answer 9

that will lead to:\[\tan(2\theta)=\infty\]

Answer 10

therefore:\[2\theta=?\]

Answer 11

∞?

Answer 12

if you know plot this hyperbola, it's easier. Look... B^2-4AC=480 (in this case), so we have a hyperbola. A=B=7, so we have \[\theta=\pi/4\].

Answer 13

which angle will give you infinity when you take its tan?

Answer 14

i don't know

Answer 15

have you studied trigonometry?

Answer 16

no, this is in my precalc course

Answer 17

hmmmm... well you do need to understad basic trigonometry in order to do this question

Answer 18

that is why i came here. i did not know how to do this type of question and came for help.

Answer 19

this might help you understand the basics: http://www.clarku.edu/~djoyce/trig/

Answer 20

there are a good set of short online videos on this here as well: http://www.khanacademy.org/math/trigonometry

Answer 21

Please, how book you use?

Answer 22

I am using the Larson Hostetler Seventh Edition Precalculus book

Answer 23

@netlopes1 , how did you get pi/4 ?

Answer 24

@dpaInc do you know how to solve this type of problem?

Answer 25

Please, read these pages....

Answer 26

yes.. but i was doing it the way @asnaseer , was doing it... eventually, you should come up with pi/4 as the rotation.
but i think there is some other way and that's why i asked @netlopes1 how he came up with that answer.... i think he can explain his method better...

Answer 27

Simply, i used this link: http://en.wikipedia.org/wiki/Rotation_of_axes and observe that A=B=7 and this is a hyperbola equation, ok? In these cases the angle is 45º or pi/4.

Answer 28

Ok, this may (or mostly likely will) get messy. However, this method does work. Let me know if it makes sense or not.



Start with the formula given on the wiki page


(A*cos^2(theta)+Bsin(theta)*cos(theta)+C*sin^2(theta))*x^2

+(A*sin^2(theta)-Bsin(theta)*cos(theta)+C*cos^2(theta))*y^2

+(D*cos(theta)+E*sin(theta))*x

+(-D*sin(theta)+E*cos(theta))*y

+F = 0


The formula is broken up into multiple lines (this is one really really big equation). The idea here is to plug in theta = pi/4 and evaluate


-------------------------------------------------------


Step 1) Plug in theta = pi/4


(7*cos^2(pi/4)+26*sin(pi/4)*cos(pi/4)+7*sin^2(pi/4))*x^2

+(7*sin^2(pi/4)-26*sin(pi/4)*cos(pi/4)+7*cos^2(pi/4))*y^2

+(0*cos(pi/4)+0*sin(pi/4))*x

+(-0*sin(pi/4)+0*cos(pi/4))*y

+24 = 0


-------------------------------------------------------


Step 2) Evaluate the sine and cosine of pi/4 (in this case, they both evaluate to 0.707106781186547 approximately)


(7*0.5+26*0.707106781186547*0.707106781186548+7*0.5)*x^2

+(7*0.5-26*0.707106781186547*0.707106781186548+7*0.5)*y^2

+(0*0.707106781186548+0*0.707106781186547)*x

+(-0*0.707106781186547+0*0.707106781186548)*y

+24 = 0

-------------------------------------------------------


Step 3) Square each value


(7*0.5+26*0.707106781186547*0.707106781186548+7*0.5)*x^2

+(7*0.5-26*0.707106781186547*0.707106781186548+7*0.5)*y^2

+(0*0.707106781186548+0*0.707106781186547)*x

+(-0*0.707106781186547+0*0.707106781186548)*y

+24 = 0


-------------------------------------------------------


Step 4) Multiply


(3.5+13+3.5)*x^2

+(3.5-13+3.5)*y^2

+(0+0)*x

+(0+0)*y

+24 = 0

-------------------------------------------------------


Step 5) Combine like terms


(20)*x^2

+(-6)*y^2

+(0)*x

+(0)*y

+24 = 0


-------------------------------------------------------


Step 6) Condense into one line (this is now possible since the equation is much shorter/smaller.


20x^2-6y^2+0x+0y+24 = 0


-------------------------------------------------------


Step 7) Erase the 0x and 0y terms


20x^2-6y^2+24 = 0


======================================================================================================================


Answer:


So 7x^2+26xy+7y^2+24=0 rotates pi/4 radians (or 45 degrees) clockwise to get 20x^2-6y^2+24 = 0


This is the equivalent of saying that rotating the entire axis system pi/4 radians (or 45 degrees) counter-clockwise will turn 7x^2+26xy+7y^2+24=0 into 20x^2-6y^2+24 = 0

Answer 29

As confirmation, you can graph the two conics and you'll see that one is a rotated version of the other.

Answer 30

But this is an example of what the answer might be...

Answer 31

ah, so I'm not done yet lol since they want the answer in parametric form

Answer 32

one second

Answer 33

ok... i'll look over the first few steps again

Answer 34

Are you sure that's the answer? I plotted the answer and I got an ellipse, but the original problem is a hyperbola....so something is off. Is there a typo in the original problem?

Answer 35

no, that is one of 5 choices... the others are

Answer 36

oh ok, thank you for providing the other answers

Answer 37

no problem, you are the one helping me :)

Answer 38

If possible, can I get a screenshot of the original problem?

Answer 39

sure... oh no! i gave the wrong other answers. hold on

Answer 40

sure thing

Answer 41

the original is
Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form.

7x^2 + 26xy + 7y^2 - 24 = 0

Answer 42

the answers are:

Answer 43

ah that makes a lot more sense, thx

Answer 44

one sec

Answer 45

so sorry for the mix up

Answer 46

no worries

Answer 47

Start with the formula given on the wiki page


(A*cos^2(theta)+Bsin(theta)*cos(theta)+C*sin^2(theta))*x^2

+(A*sin^2(theta)-Bsin(theta)*cos(theta)+C*cos^2(theta))*y^2

+(D*cos(theta)+E*sin(theta))*x

+(-D*sin(theta)+E*cos(theta))*y

+F = 0


The formula is broken up into multiple lines (this is one really really big equation). The idea here is to plug in theta = pi/4 and evaluate



-------------------------------------------------------

Step 1) Plug in theta = pi/4


(7*cos^2(pi/4)+26*sin(pi/4)*cos(pi/4)+7*sin^2(pi/4))*x^2

+ (7*sin^2(pi/4)-26*sin(pi/4)*cos(pi/4)+7*cos^2(pi/4))*y^2

+ (0*cos(pi/4)+0*sin(pi/4))*x

+ (-0*sin(pi/4)+0*cos(pi/4))*y

+ -24 = 0


-------------------------------------------------------

Step 2) Evaluate the sine and cosine of pi/4 (in this case, they both evaluate to 0.707106781186547 approximately)


(7*0.5+26*0.707106781186547*0.707106781186548+7*0.5)*x^2

+(7*0.5-26*0.707106781186547*0.707106781186548+7*0.5)*y^2

+(0*0.707106781186548+0*0.707106781186547)*x

+(-0*0.707106781186547+0*0.707106781186548)*y

+ -24 = 0

-------------------------------------------------------

Step 3) Square each value


(7*0.5+26*0.707106781186547*0.707106781186548+7*0.5)*x^2

+(7*0.5-26*0.707106781186547*0.707106781186548+7*0.5)*y^2

+(0*0.707106781186548+0*0.707106781186547)*x

+(-0*0.707106781186547+0*0.707106781186548)*y

+ -24 = 0


-------------------------------------------------------

Step 4) Multiply

(3.5+13+3.5)*x^2

+(3.5-13+3.5)*y^2

+(0+0)*x

+(0+0)*y

+ -24 = 0

-------------------------------------------------------

Step 5) Combine like terms


(20)*x^2

+(-6)*y^2

+(0)*x

+(0)*y

+ -24 = 0


-------------------------------------------------------


Step 6) Condense into one line (this is now possible since the equation is much shorter/smaller.


(20)*x^2+(-6)*y^2+(0)*x+(0)*y+ -24 = 0


-------------------------------------------------------


Step 7) Simplify


20x^2 - 6y^2 - 24 = 0


-------------------------------------------------------


Step 8) Add 24 to both sides


20x^2 - 6y^2 = 24


-------------------------------------------------------


Step 9) Divide every term by 24


20x^2/24 - 6y^2/24 = 24/24

-------------------------------------------------------


Step 10) Simplify and rearrange terms


x^2/(6/5) - y^2/4 = 1


======================================================================================================================


Answer:


So the final answer is \[\Large \frac{x^2}{\frac{6}{5}} - \frac{y^2}{4} = 1\]


Note: I'm using x in place of x', but they really are the same thing (one is just the converted/rotated coordinate)

Answer 48

So the answer using x' and y' is \[\Large \frac{\left(x^{\prime}\right)^2}{\frac{6}{5}} - \frac{\left(y^{\prime}\right)^2}{4} = 1\]

Answer 49

THANK YOU!!!! no one else was really helpful... i learn much easier by seeing an example of a problem done than just being given a formula.

Answer 50

You're very welcome, I recommend practicing with this idea a lot more because there are a ton of steps which are very very long.

Answer 51

i definitely will. i have 4 other similar questions to figure out and really needed step by step instructions.

Answer 52

oh and I'd go over the wiki page in more depth if you're not sure how the formula works and such

Answer 53

Thanks

Answer 54

sure thing