Question

Three lines $x+2y+3=0,x+2y–7=0$ and $2x–y–4=0$ form $3$ sides of two squares. Find the equation of remaining sides of these squares.

Answer 1

Distance between the two parallel lines is $\frac{|7+3|}{\sqrt{5}}=2\sqrt{5}.$

The equations of sides $A$ and $C$ are of the form $2x–y+k=0$.

Since distance between sides $A$ and $B=$ distance between sides $B$ and $C$ hence

$\frac{|k-(-4)|}{\sqrt{5}}=2\sqrt{5}$

$\Rightarrow \frac{k+4}{\sqrt{5}}=\pm 2\sqrt{5}$

$\Rightarrow k+4=\pm 2\times 5=\pm 10$

$\Rightarrow k=-4+10,-4-10$

$\therefore k=6,-14$

Hence the fourth sides of the two squares are

$\left(i\right)2x–y+6=0,\left(ii\right)2x–y–14=0$