Question

Find the equations to the diagonals of the rectangle the equations of whose sides are x = a, x = a', y = b and y = b'.

Answer 1

The rectangles formed by the lines x = a, x = a', y = b and y = b' is shown below:

Clearly, the vertices of the rectangle are $A\left(a,b\right),B\left(a^{\text{'}},b\right),C\left(a^{\text{'}},b^{\text{'}}\right)\mathrm{and}D\left(a,b^{\text{'}}\right)$.

The diagonal passing through $A\left(a,b\right)\mathrm{and}C\left(a^{\text{'}},b^{\text{'}}\right)$ is

$$\begin{gathered} y-b=\frac{b^{\text{'}}-b}{a^{\text{'}}-a}\left(x-a\right) \\ \Rightarrow \left(a^{\text{'}}-a\right)y-b\left(a^{\text{'}}-a\right)=\left(b^{\text{'}}-b\right)x-a\left(b^{\text{'}}-b\right) \\ \Rightarrow \left(a^{\text{'}}-a\right)y-\left(b^{\text{'}}-b\right)x=-a\left(b^{\text{'}}-b\right)+b\left(a^{\text{'}}-a\right) \\ \Rightarrow \left(a^{\text{'}}-a\right)y-\left(b^{\text{'}}-b\right)x=b{a}^{\text{'}}-a{b}^{\text{'}} \end{gathered}$$

And, the diagonal passing through $B\left(a^{\text{'}},b\right)\mathrm{and}D\left(a,b^{\text{'}}\right)$ is

$$\begin{gathered} y-b=\frac{b^{\text{'}}-b}{a-a^{\text{'}}}\left(x-a^{\text{'}}\right) \\ \Rightarrow \left(a-a^{\text{'}}\right)y-b\left(a-a^{\text{'}}\right)=\left(b^{\text{'}}-b\right)x-a^{\text{'}}\left(b^{\text{'}}-b\right) \\ \Rightarrow \left(a-a^{\text{'}}\right)y-\left(b^{\text{'}}-b\right)x=-a^{\text{'}}\left(b^{\text{'}}-b\right)+b\left(a-a^{\text{'}}\right) \\ \Rightarrow \left(a^{\text{'}}-a\right)y+\left(b^{\text{'}}-b\right)x=a^{\text{'}}{b}^{\text{'}}-ab \end{gathered}$$

Hence, the equations of the diagonals are $\left(a^{\text{'}}-a\right)y-\left(b^{\text{'}}-b\right)x=b{a}^{\text{'}}-a{b}^{\text{'}}$ and $\left(a^{\text{'}}-a\right)y+\left(b^{\text{'}}-b\right)x=a^{\text{'}}{b}^{\text{'}}-ab$.