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.