Problem:
Suppose a line $L$ is given by the equation $\frac{x}{a} + \frac{y}{b}=1$, where $a$ and $b$ are non-zero real numbers. Let $\Re_{\frac{\pi}{2}}$ be the counterclockwise rotation of the plane by $90$ $degrees$ with the center at $(0,0)$. Find an equation of $\Re{\frac{\pi}{2}}(L)$ in the form $\frac{x}{u} + \frac{y}{v} = 1$.
I know that a $90$ $degree$ rotation means $(x,y)\mapsto(-y,x).$
My Solution:
Suppose we have $\frac{x}{a}+\frac{y}{b}=1$. If we multiply through by $a$ we have $x+\frac{ya}{b}=a$ and then multiplying through by $b$ gives us $xb+ya=ab$. Then subtracting $xb$ gives $ya=-xb+ab$. Finally, dividing by $a$ yields $y=\frac{ab-bx}{a}$.
I'm not sure if I have done this correctly and I was hoping someone could look over my work?
If you are not convinced, try this more analytical approach:
Let $\theta$ be the original slope angle of the line. If you think of the line in $y=mx + c$ form, then $m=tan(\theta)$. Now, when you rotate the plane, the new slope angle is $90+\theta$ (Draw the picture). So the new slope is $m^*=tan(90+\theta)=-cot(\theta)=-1/m$. Also, from the picture, you can also deduce that the new intercept $c^*$ satisfies $tan(\theta)=c/c^*$. Hence you get $c^*=c/m$. Your new equation is $y=m^*x+c^*$. Substitute the values of $m^*$ and $c^*$. Then further substitute the values of $m$ and $c$ in terms of $a$ and $b$. Finally rearrange to get in the form you want.