Question

If 2x + y – 5 = 0 and 4x + 2y – 15 = 0 are two parallel sides of a square, then its area is ____________.

Answer 1

Since 2x + y − 5 = 0 and 4x + 2y − 15 = 0 are parallel
i.e. 2x + y − 5 = 0
2x + y − $\frac{15}{2}$ = 0 are parallel
Distance between these parallel lines is

$\begin{array}{rcl}\frac{\left|{\displaystyle 5-\frac{15}{2}}\right|}{\sqrt{4+1}}& =& \frac{\left|10-15\right|}{2\sqrt{5}}=\frac{5}{2\sqrt{5}}\\ & =& \frac{\sqrt{5}}{2}\\ & & \end{array}$
i.e. Distance between parallel sides which is length of side of square
$\therefore \mathrm{Area}\mathrm{of}\mathrm{square}\mathrm{is}{\left(\frac{\sqrt{5}}{2}\right)}^2=\frac{5}{4}\mathrm{sq}.\mathrm{units}$