# Combined Equation of pair of Lines – 01

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### Type – I A: To Find Joint Equation When Separate Equations are Given:

#### ALGORITHM :

1. Write equations of lines in the form u = 0 and  v = 0.
2. Find u.v = 0.
3. Simplify the L.H.S. of the joint equation.

#### Example – 1:

• Obtain the joint equation of the co-ordinate axes.
• Solution:

Equation of the x-axis is  y =  0 .

Equation of the y-axis is  x =  0

The joint equation of the two co-ordinate axis is x y = 0

#### Example – 2:

• Find the joint equation of the two lines whose separate equations are 3x -2y = 0 and 4x + y = 0.
• Solution:

Equation of the first line is 3x -2y = 0

Equation of the second line is  4x + y= 0

Thus the joint equation of a line is given by

(3x – 2y)(4x + y) = 0

∴ 3x (4x + y)   – 2y (4x + y) = 0

∴ 12x² + 3xy – 8xy – 2y² = 0

∴ 12x² – 5xy – 2y² = 0

This is the required combined equation.

#### Example – 3:

• Find the combined equation of the two lines whose separate equations are 3x + 4y = 0 and 2x = 3y
• Solution:

Equation of the first line is 3x + 4y = 0

Equation of the second line is  2x = 3y i.e 2x – 3y = 0

Thus the joint equation of a line is given by

(3x + 4y)(2x – 3y ) = 0

∴ 3x (2x – 3y)   + 4y (2x – 3y ) = 0

∴ 6x² – 9xy + 8xy – 12y² = 0

∴  6x² – xy – 12y² = 0

This is the required combined equation.

#### Example 4:

• Find the combined equation of the lines whose separate equations are 2x+y = 0 and 3x-5y = 0.
• Solution:

Given equations of lines are 2x+y = 0 and 3x-5y = 0

(2x+y )(3x-5y) = 0

∴ 2x(3x-5y) + y(3x-5y) = 0

∴ 6x² – 10xy + 3xy  – 5y² = 0

∴  6x² – 7xy  – 5y² = 0

This is the required combined equation.

#### Example 5:

• Find the combined equation of the lines whose separate equations are 3x – 2y + 1 = 0 and 4x – 3y + 5 = 0
• Solution:

Given equations of lines are 3x – 2y + 1 = 0 and 4x – 3y + 5 = 0

(3x – 2y + 1)(4x – 3y + 5) = 0

3x(4x – 3y + 5) – 2y(4x – 3y + 5) + 1(4x – 3y + 5) =

∴     12x² – 9xy + 15x – 8xy + 6y² – 10y + 4x – 3y + 5 = 0

∴      12x² – 17 xy  + 6y² + 19x  – 13y + 5 = 0

This is the required combined equation.

#### Example 6:

• Find the combined equation of the lines whose separate equations are x + y = 3 and  2x + y – 1 = 0.
• Solution:

Given equations of lines are x + y – 3 = 0 and  2x + y – 1 = 0

(x + y – 3)(2x + y – 1) = 0

x(2x + y – 1) + y(2x + y – 1) – 3(2x + y – 1) = 0

∴   2x² + xy – x + 2xy + y² – y – 6x – 3y + 3 = 0

∴    2x² + 3xy  + y² – 7x  – 4y + 3 = 0

This is the required combined equation.

#### Example 7:

• Find the combined equation of the lines whose separate equations are 3x + 2y -1 = 0 and x + 3y -2 = 0.
• Solution:

Given equations of lines are 3x + 2y -1 = 0 and x + 3y -2 = 0.

(3x + 2y -1 )(x + 3y -2) = 0

3x(x + 3y -2) + 2y(x + 3y -2) – 1(x + 3y -2) = 0

∴   3x² + 9xy – 6x + 2xy + 6y² – 4y – x – 3y + 2 = 0

∴   3x² + 11xy  + 6y² – 7x  – 7y + 2 = 0

This is the required combined equation.

#### Example 8:

• Find the combined equation of the lines whose separate equations are 2x+y = 0 and 3x-5y = 0.
• Solution:

Given equations of lines are 2x+y = 0 and 3x-5y = 0

(2x+y )(3x-5y) = 0

∴ 2x(3x-5y) + y(3x-5y) = 0

∴ 6x² – 10xy + 3xy  – 5y² = 0

∴  6x² – 7xy  – 5y² = 0

This is the required combined equation.

#### Example 9:

• Find the combined equation of the lines whose separate equations are x + 2y – 1 = 0 and 2x – 3y + 2 = 0.
• Solution:

Given equations of lines are x + 2y – 1 = 0 and 2x – 3y + 2 = 0.

(x + 2y – 1 )(2x – 3y + 2) = 0

∴  x(2x – 3y + 2) + 2y(2x – 3y + 2) – 1(2x – 3y + 2) = 0

∴ 2x² – 3xy + 2x + 4xy – 6y² + 4y – 2x + 3y – 2 = 0

∴  2x² + xy  – 6y² + 7y – 2 = 0

This is the required combined equation.

### Type – IB To Find Joint Equation When Angles Made by the Lines with  X-axis or Y-axis are Given:

#### ALGORITHM :

1. If θ is the inclination of a line, then its slope is given by m = tan θ.
2. Find slopes m1 and m2 of the two lines.
3. Use y = mx form to find equations of the two lines.
4. Write equations of lines in the form u = 0 and  v = 0.
5. Find u.v = 0.
6. Simplify the L.H.S. of the joint equation.

#### Example 10:

• Find the combined equation of the lines passing through the origin and making an angle of 30° with positive direction of x-axis
• Solution:

Slope of the first line = m1 = tan 30° =  1/3

The equation the first line is y = m1 x

y = 1/3x

3 y  = x

x – √3 y  = 0     …………. (1)

Slope of the second line = m2 = tan (-30°)= – tan 30°  =  -1/3

The equation the second line is y = m2 x

y =-  1/3x

3 y  = – x

x + √3 y  = 0     …………. (2)

Their joint equation is (x – √3 y) (x + √3 y) = 0

x² – 3y² = 0

This is the required combined equation.

#### Example 11 :

• Find the combined equation of the lines passing through the origin and making an angle of 45° with positive direction of x – axis
• Solution:

Slope of the first line = m1 = tan 45° =  1

The equation the first line is y = m1 x

y =  1.x

x – y  = 0     …………. (1)

Slope of the second line = m2 = tan (-45°)= – tan 45°  =  -1

The equation the second line is y = m2 x

y =-  1 x

x +  y  = 0     …………. (2)

Their joint equation is (x – y) (x + y) = 0

x² – y² = 0

This is the required combined equation.

#### Example 12:

• Find the combined equation of the lines passing through the origin and making an angle of 60O with positive direction of x – axis
• Solution:

Slope of the first line = m1 = tan 60° =  3

The equation the first line is y = m1 x

y = 3x

3 x  – y = 0     …………. (1)

Slope of the second line = m2 = tan (-60°)= – tan 60°  =  –3

The equation the second line is y = m2 x

y =-  3x

3 x  + y = 0     …………. (2)

Their joint equation is (3 x  – y) (3 x + y) = 0

3x² – y² = 0

This is the required combined equation.

#### Example 13:

• Find the combined equation of the lines passing through the origin and making an angle of 60O with y – axis
• Solution:

Slope of the first line = m1 = tan 30° =  1/3

The equation the first line is y = m1 x

y = 1/3x

3 y  = x

x – √3 y  = 0     …………. (1)

Slope of the second line = m2 = tan (150°)= tan(180° – 30°) = -tan 30°  =  -1/3

The equation the second line is y = m2 x

y =-  1/3x

3 y  = – x

x + √3 y  = 0     …………. (2)

Their joint equation is (x – √3 y) (x + √3 y) = 0

x² – 3y² = 0

This is the required combined equation.

#### Example 14:

• Find the joint equation of a pair of lines through the origin having and having inclination 30° and 150°.
• Solution:

Slope of the first line = m1 = tan 30° =  1/3

The equation the first line is y = m1 x

y = 1/3x

3 y  = x

x – √3 y  = 0     …………. (1)

Slope of the second line = m2 = tan (150°)= tan(180° – 30°) = -tan 30°  =  -1/3

The equation the second line is y = m2 x

y =-  1/3x

3 y  = – x

x + √3 y  = 0     …………. (2)

Their joint equation is (x – √3 y) (x + √3 y) = 0

x² – 3y² = 0

This is the required combined equation.

#### Example 15:

• Find the joint equation of a pair of lines through the origin having and having inclination 60° and 120°
• Solution:

Slope of the first line = m1 = tan 60° =  3

The equation the first line is y = m1 x

y = 3x

3 x  – y = 0     …………. (1)

Slope of the second line = m2 = tan (120°)= tan (180° – 60°) = – tan 60°  =  –3

The equation the second line is y = m2 x

y =-  3x

3 x  + y = 0     …………. (2)

Their joint equation is (3 x  – y) (3 x + y) = 0

3x² – y² = 0

This is the required combined equation.

#### Example 16:

Find the joint equation of a pair of lines through the origin having and having inclination π/3 and 5π/3.

Solution:

Slope of the first line = m1 = tan π/3 =  3

The equation the first line is y = m1 x

y = 3x

3 x  – y = 0     …………. (1)

Slope of the second line = m2 = tan (5π/3)= tan (2π – π/3) = – tan π/3  =  –3

The equation the second line is y = m2 x

y =-  3x

3 x  + y = 0     …………. (2)

Their joint equation is (3 x  – y) (3 x + y) = 0

3x² – y² = 0

This is the required combined equation.

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