Question

The diagonals of the parallelogram whose sides are $\mathrm{lx}+\mathrm{my}+\mathrm{n}=0,\mathrm{lx}+\mathrm{my}+\mathrm{n}\text{'}=0,\mathrm{mx}+\mathrm{ly}+\mathrm{n}=0,\mathrm{mx}+\mathrm{ly}+\mathrm{n}\text{'}=0$ includes an angle

• A. $\frac{\mathrm{\pi }}{2}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. None of these

Answer 1

The correct option is A

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option:

Determining the angles of the parallelogram.

Given sides of the parallelogram are

$$\begin{gathered} \mathrm{lx}+\mathrm{my}+\mathrm{n}=0\dots \left(\mathrm{i}\right) \\ \mathrm{lx}+\mathrm{my}+\mathrm{n}\text{'}=0\dots \left(\mathrm{ii}\right) \\ \mathrm{mx}+\mathrm{ly}+\mathrm{n}=0\dots \left(\mathrm{iii}\right) \\ \mathrm{mx}+\mathrm{ly}+\mathrm{n}\text{'}=0\dots \left(\mathrm{iv}\right) \end{gathered}$$

Slope of line $\left(\mathrm{i}\right)=-\frac{\mathrm{l}}{\mathrm{m}}$

Slope of line $\left(\mathrm{ii}\right)=-\frac{\mathrm{l}}{\mathrm{m}}$

Slope of line $\left(\mathrm{iii}\right)=-\frac{\mathrm{m}}{\mathrm{l}}$

Slope of line $\left(\mathrm{iii}\right)=-\frac{\mathrm{m}}{\mathrm{l}}$

So, $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$ are parallel lines and, $\left(\mathrm{iii}\right)$ and $\left(\mathrm{iv}\right)$ are parallel lines, since slopes of two lines are equal when they are parallel.

Distance between two parallel lines is given as $d=\left|\frac{\left(c_2-c_1\right)}{\sqrt{a^2+b^2}}\right|$

Hence, distance between $\left(\mathrm{i}\right)$ and $\left(\mathrm{ii}\right)$ $=\left|\frac{\left(\mathrm{n}\text{'}-\mathrm{n}\right)}{\sqrt{{\mathrm{l}}^2+{\mathrm{m}}^2}}\right|\dots \left(\mathrm{v}\right)$

Distance between $\left(\mathrm{iii}\right)$ and $\left(\mathrm{iv}\right)$ $=\left|\frac{\left(\mathrm{n}\text{'}-\mathrm{n}\right)}{\sqrt{{\mathrm{l}}^2+{\mathrm{m}}^2}}\right|\dots \left(\mathrm{vi}\right)$

Here, we can note that $\left(\mathrm{v}\right)=\left(\mathrm{vi}\right)$

therefore, the distance between the parallel sides is equal and the parallelogram is a rhombus.

In a rhombus, the diagonals bisect each other at $90°$ So, the angle between the diagonals is $\frac{\mathrm{\pi }}{2}$.

Hence, option (A) is correct.