I have done it now. The second part is done by starting similarly to the first part.
Yes. To do the first part, look at .
So x'= 5x and y'= -4x. x= x'/5 so y'= -4(x'/5)= (-4/5)x'.
Any "line through the origin" is of the form y= ax with slope a. The line at right angles to that is y= (-1/a)x so you want to find a such that
That is, we must have x'= (2+a)x and y'= (-1/a)x'= (2- 2a)x. Dividing one equation by the other so that we eliminate x' on both sides of the equation, . Multiply on both sides by (2+ 2a)a to get rid of the fractions.
which reduces to .
Ouch, I divided the two equations incorrectly. Dividing x'= (2+ a)x by (-1/a)x'= (2- 2a)x gives or . Multiplying both sides by 2- 2a gives -(2- 2a)a= -2a+ 2a^2= 2+ a. which has roots a= 2 and a= -1/2. the lines, y= ax, are y= 2x and y= (-1/2)x as you said.