a)
The rotation transformations are:
[tex] x=x' \cos \theta - y' \sin \theta\\
y= x' \sin \theta + y' \cos \theta[/tex]
where $(x', y')$ are new coordinates.
Since we want an anti-clockwise rotation of the axes, put $\theta= 90$° to get the required coordinates as $(4,-1)$
b)
The equation of the given line is $x-y=0$
Use the mirror formula:
[tex] x'= x_o - \frac{2 a}{a^2+ b^2}(a x_o + by_o +c)[/tex]
Replace $x$ by $y$ above to get the corresponding formula for the y-coordinate.
Here we have
$a=1$
$b=-1$
$c=0$
$x_o= 1$
$y_o =4$
Substitute these values to get the required coordinates as $(4,1)$
c)
$(5,3)$ maps onto $(1,1)$
Let's keep it simple here: To get the new coordinates you subtract 4 from the x coordinate and 2 from the y coordinate.
Thus for our point $(1,4)$ our image is $(1-4, 4-2)$ which is $(-3,2)$
