We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
149
4
avatar+895 

Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.

\(x^2+4xy+y^2-3=0\)

 

\(x^2-4xy+y^2-3=0\)

 Jun 17, 2019
 #1
avatar+28176 
+3

Here is the first equation:

 

I'll leave you to use the same technique on the second equation.

 Jun 17, 2019
 #2
avatar+104688 
+1

x^2 - 4xy + y^2 - 3  = 0

 

We have the form  Ax^2 + Bxy  + Cy^2 + F = 0

 

The  angle of rotation is given by

 

cot (2θ)  =  A - C       =      1  - 1

                 _____             _____  =      0

                   B                      -4

 

arccot (0)  = 2θ

 

pi/2  = 2θ

 

pi/4  = θ =  45°

 

Let  x  = x'cos(45) - y'sin(45)   =  [1/√2] [x' - y']

Let y  = x'sin(45) + y'cos(45) = [ 1/√2] [ x' +y']

 

Sub this into  x^2 - 4xy + y^2 - 3  = 0

 

( [1/√2] [x' - y'])^2  - 4  [1/√2] [x' - y'][ 1/√2] [ x' +y'] + ([ 1/√2] [ x' +y'])^2 - 3  = 0

 

(1/2)[x'^2 - 2x'y' + y'^2] - 4(1/2)[x'^2 -y'^2] +(1/2)[x'^2 + 2x'y' + y'^2] - 3 = 0

 

x'^2 + y'^2 - 2x'^2 + 2y'^2  - 3  = 0

 

3y'^2 - x'^2  =  3

 

y'^2 -  x'^2/3  = 1

 

So...this is the graph  of    y^2 - x^2/3  = 1    rotated 45°

 

See the graph and its rotation here : https://www.desmos.com/calculator/8lzzx1eflt

 

 

cool cool cool

 Jun 17, 2019
edited by CPhill  Jun 18, 2019
 #3
avatar+895 
+1

Thank you so much CPhill. This was actually the way my textbook wants me to do it so helps a lot.

AdamTaurus  Jun 17, 2019
 #4
avatar+104688 
0

OK....glad to help.....

 

cool cool cool

CPhill  Jun 18, 2019

45 Online Users

avatar
avatar