Question 1: Prove that the area of the parallelogram formed by the lines:

is

Deduce the condition for these lines to form a rhombus.

Answer:

Let be a parallelogram the equation of whose sides are given as:

Let be the distances between the pair of parallel sides of the parallelogram.

In we get,

Therefore, the area of a parallelogram is , where

Let be the slopes of sides respectively. Then,

The angle between is given by

We have

and

In case of a rhombus, are equal. Hence the condition is:

Question 2: Prove that the area of the parallelogram formed by the lines is sq. units.

Answer:

Given lines are:

… … … … … i)

… … … … … ii)

… … … … … iii)

… … … … … iv)

sq. units

Question 3: Show that the diagonals of the parallelogram whose sides are include an angle .

Answer:

Given equations:

… … … … … i)

… … … … … iii)

… … … … … ii)

… … … … … iv)

Since .

Therefore the diagonals are perpendicular to each other.