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# Class 11 RD Sharma Solutions- Chapter 23 The Straight Lines- Exercise 23.17

• Last Updated : 19 Jan, 2021

### Deduce the condition for these lines to form a rhombus.

Solution:

Given:

The given lines are

a1x + b1y + c1 = 0 —(equation-1)

a1x + b1y + d1 = 0 —(equation-2)

a2x + b2y + c2 = 0 —(equation-3)

a2x + b2y + d2 = 0 —(equation-4)

Let us prove, the area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a1x + b1y + d1 = 0, a2x + b2y + c2 = 0, a2x + b2y + d2 = 0 is

sq. units.

The area of the parallelogram formed by the lines a1x + b1y + c1 = 0, a1x + b1y + d1 = 0, a2x + b2y + c2 = 0 and a2x + b2y + d2 = 0 is given below:

Area  =

Since, \begin{vmatrix}a_1&a_2\\b_1&b_2\end{vmatrix} = a1b2 – a2b1

Therefore, Area =

If the given parallelogram is a rhombus, then the distance between the pair of parallel lines is equal.

Therefore,

Hence proved.

### Question 2. Prove that the area of the parallelogram formed by the lines 3x â€“ 4y + a = 0, 3x â€“ 4y + 3a = 0, 4x â€“ 3y â€“ a = 0 and 4x â€“ 3y â€“ 2a = 0 is  sq. units.

Solution:

Given:

The given lines are

3x âˆ’ 4y + a = 0 —(equation-1)

3x âˆ’ 4y + 3a = 0 —(equation-2)

4x âˆ’ 3y âˆ’ a = 0 —(equation-3)

4x âˆ’ 3y âˆ’ 2a = 0 —(equation-4)

We have to prove, the area of the parallelogram formed by the lines 3x â€“ 4y + a = 0, 3x â€“ 4y + 3a = 0, 4x â€“ 3y â€“ a = 0 and 4x â€“ 3y â€“ 2a = 0 is  sq. units.

From above solution, we know that

Area of the parallelogram =

Area of the parallelogram =  sq. units

Hence proved.

### Question 3. Show that the diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + nâ€™ = 0, mx + ly + n = 0 and mx + ly + nâ€™ = 0 include an angle Ï€/2.

Solution:

Given:

The given lines are

lx + my + n = 0 —(equation-1)

mx + ly + nâ€™ = 0 —(equation-2)

lx + my + nâ€™ = 0 —(equation-3)

mx + ly + n = 0 —(equation-4)

We have to prove, the diagonals of the parallelogram whose sides are lx + my + n = 0, lx + my + nâ€™ = 0, mx + ly + n = 0 and mx + ly + nâ€™ = 0 include an angle .

By solving equation (1) and (2), we will get

B =

Solving equation (2) and (3), we get

C =

Solving equation (3) and (4), we get

D =

Solving equation (1) and (4), we get

A =

Let m1 and m2 be the slope of AC and BD.

Now,

m1

m2

Therefore,

m1m2 = -1

Hence proved.

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