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

is

sq. units. Deduce the condition for these lines to form a rhombus.

Answer:

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

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

In and we get,

  and  respectively.

  and

Now, Area of parallelogram        

Also, Area of parallelogram        

Therefore, the area of a parallelogram is ,  where and are the distances between pairs of parallel sides and is the angle between the two adjacent sides.

Let and be the slopes of sides and respectively. Then,

  and

The angle between and is given by

We have Distance between parallel sides and

and Distance between parallel sides and

Therefore are of parallelogram

In case of a rhombus, and 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)

Area of parallelogram sq. units

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

Answer:

Given equations:

     … … … … … i)

     … … … … … iii)

     … … … … … ii)

     … … … … … iv)

Solving i) and ii) we get the point of intersection

Solving ii) and iii) we get the point of intersection

Solving iii) and iv) we get the point of intersection

Solving iv) and i) we get the point of intersection

Slope of

Slope of

Since .

Therefore the diagonals are perpendicular to each other.

Hence the diagonals include an angle