Mathematics OpenStudy (anonymous):

Rotate the axes to eliminate the xy-term in the equation. Then write the equation in standard form. 7x2 + 26xy + 7y2 - 24 = 0 OpenStudy (asnaseer): OpenStudy (anonymous):

I've looked at that, but I'm not sure how to apply it to my question. OpenStudy (asnaseer):

If you read through that web page you'll find it actually shows you the answer to your question. OpenStudy (asnaseer):

Look at the section titled: Elimination of the xy term by the rotation formula OpenStudy (anonymous):

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. OpenStudy (asnaseer):

which part do you not understand? OpenStudy (anonymous):

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 OpenStudy (asnaseer):

A-C=0 is the correct approach OpenStudy (asnaseer):

that will lead to:$\tan(2\theta)=\infty$ OpenStudy (asnaseer):

therefore:$2\theta=?$ OpenStudy (anonymous):

∞? OpenStudy (netlopes1):

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 $\theta=\pi/4$. OpenStudy (asnaseer):

which angle will give you infinity when you take its tan? OpenStudy (anonymous):

i don't know OpenStudy (asnaseer):

have you studied trigonometry? OpenStudy (anonymous):

no, this is in my precalc course OpenStudy (asnaseer):

hmmmm... well you do need to understad basic trigonometry in order to do this question OpenStudy (anonymous):

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

there are a good set of short online videos on this here as well: http://www.khanacademy.org/math/trigonometry OpenStudy (netlopes1): OpenStudy (anonymous):

I am using the Larson Hostetler Seventh Edition Precalculus book OpenStudy (anonymous):

@netlopes1 , how did you get pi/4 ? OpenStudy (anonymous):

@dpaInc do you know how to solve this type of problem? OpenStudy (netlopes1): OpenStudy (anonymous):

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... OpenStudy (netlopes1):

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. jimthompson5910 (jim_thompson5910):

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 jimthompson5910 (jim_thompson5910):

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

But this is an example of what the answer might be... jimthompson5910 (jim_thompson5910):

ah, so I'm not done yet lol since they want the answer in parametric form jimthompson5910 (jim_thompson5910):

one second OpenStudy (anonymous):

ok... i'll look over the first few steps again jimthompson5910 (jim_thompson5910):

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? OpenStudy (anonymous):

no, that is one of 5 choices... the others are jimthompson5910 (jim_thompson5910):

oh ok, thank you for providing the other answers OpenStudy (anonymous):

no problem, you are the one helping me :) jimthompson5910 (jim_thompson5910):

If possible, can I get a screenshot of the original problem? OpenStudy (anonymous):

sure... oh no! i gave the wrong other answers. hold on jimthompson5910 (jim_thompson5910):

sure thing OpenStudy (anonymous):

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 OpenStudy (anonymous): jimthompson5910 (jim_thompson5910):

ah that makes a lot more sense, thx jimthompson5910 (jim_thompson5910):

one sec OpenStudy (anonymous):

so sorry for the mix up jimthompson5910 (jim_thompson5910):

no worries jimthompson5910 (jim_thompson5910):

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) jimthompson5910 (jim_thompson5910):

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

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. jimthompson5910 (jim_thompson5910):

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. OpenStudy (anonymous):

i definitely will. i have 4 other similar questions to figure out and really needed step by step instructions. jimthompson5910 (jim_thompson5910):

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

Thanks jimthompson5910 (jim_thompson5910):

sure thing

Latest Questions Vocaloid: [my art] new twich emote. light concrit welcome.
47 minutes ago 1 Reply 0 Medals Camachojenny12: small company has 20 employees. Six of these employees will be selected randomly to be interviewed as part of an employee satisfaction program.
1 hour ago 0 Replies 0 Medals answerplsplsplspls: Lily is a botanist who works for a garden that many tourists visit. The function f(s) = 2s + 30 represents the number of flowers that bloomed, where s is th
2 hours ago 0 Replies 0 Medals Annamarie: Identity Shaped by Conflict Summarize the conflict Marcus is facing.
2 hours ago 0 Replies 0 Medals IPhoneDude120: Who here misses Alaska
2 hours ago 65 Replies 0 Medals myiaxo: List at least two technology tools that can help with calculating future value of an investment.
4 hours ago 0 Replies 0 Medals myiaxo: Evan opens a savings account with $5,000. He deposits$75 every month into the account that compounds annually and has a 0.
4 hours ago 0 Replies 0 Medals 56789123: During a treasure hunt Cody walks 55.0 m south and 7.50 m to the west. What singl
4 hours ago 0 Replies 0 Medals Jasmine2123: A scientist calculated the energy required to break bonds in two unknown compounds, A and B.
4 hours ago 0 Replies 0 Medals Glupo: What is the mass (in grams) of 5.00 L of propane vapor (Cu2083Hu2088) at STP
4 hours ago 0 Replies 0 Medals