Find the equation of the invariant line through the origin

chwala
Gold Member
Messages
2,825
Reaction score
413
Homework Statement
My interest is on highlighted in yellow. part b
Relevant Equations
see attached
1712742976617.png


My approach - i think similar to ms approach.

The required Equation will be in the form ##y=mx##

##\begin{pmatrix}
a & b^2 \\
c^2 & a
\end{pmatrix} ⋅
\begin{pmatrix}
k \\
mk
\end{pmatrix} =
\begin{pmatrix}
x \\
y
\end{pmatrix}
##



##ak+b^2mk=x##
##kc^2+amk=y##

##x=k(a+b^2m)##
##k=\dfrac{x}{a+b^2m}##

##y= k(c^2+am)##
##y=\dfrac{c^2+am}{a+b^2m}x##

##m=\dfrac{c^2+am}{a+b^2m}##

##am+b^2m^2=c^2+am##
##b^2m^2-c^2=0##
##m=\sqrt {\dfrac{c^2}{b^2}}##

##m_1 = \dfrac{c}{b}## and ##m_2 = -\dfrac {c}{b}##

##y=\dfrac{c}{b}x##

and

##y=-\dfrac{c}{b}x##

Ms approach,
1712743125853.png




Any insight welcome guys!
 
Last edited:
Physics news on Phys.org
There's a general form for a matrix describing a rotation about the origin by an a gle ## \theta##.
You can derive it by seeing what happens when you rotate the point ##P=(cost, sint)## to the point ##P'=(cos(t+\theta), sin(t+\theta))##. Then expand the latter expression using the formulas for sin, cos of the sum of angles (show the map is linear), and use it to describe the matrix that takes you from ##P## to ##P'##.
Use that general form to test against the matrix you're given.
Can you take it from there?
 
WWGD said:
There's a general form for a matrix describing a rotation about the origin by an a gle ## \theta##.
You can derive it by seeing what happens when you rotate the point ##P=(cost, sint)## to the point ##P'=(cos(t+\theta), sin(t+\theta))##. Then expand the latter expression using the formulas for sin, cos of the sum of angles (show the map is linear), and use it to describe the matrix that takes you from ##P## to ##P'##.
Use that general form to test against the matrix you're given.
Can you take it from there?
I will need to check on this- self studying... thanks.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top