Find the equation of the invariant line through the origin

[chwala]

Homework Statement

My interest is on highlighted in yellow. part b

Relevant Equations

see attached

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,

Any insight welcome guys!

 

[WWGD]

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?

 

[chwala]

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.