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Science > Mathematics > Pair of Straight Lines > You are Here |

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

**ALGORITHM :**

- Write equations of lines in the form u = 0 and v = 0.
- Find u.v = 0.
- 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 :**

- If θ is the inclination of a line, then its slope is given by m = tan θ.
- Find slopes m
_{1}and m_{2}of the two lines. - Use y = mx form to find equations of the two lines.
- Write equations of lines in the form u = 0 and v = 0.
- Find u.v = 0.
- 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 = m_{1} = tan 30° = 1/√3

The equation the first line is y = m_{1} x

y = 1/√3x

√3 y = x

x – √3 y = 0 …………. (1)

Slope of the second line = m_{2} = tan (-30°)= – tan 30° = -1/√3

The equation the second line is y = m_{2} 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 = m_{1} = tan 45° = 1

The equation the first line is y = m_{1} x

y = 1.x

x – y = 0 …………. (1)

Slope of the second line = m_{2} = tan (-45°)= – tan 45° = -1

The equation the second line is y = m_{2} 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 = m_{1} = tan 60° = √3

The equation the first line is y = m_{1} x

y = √3x

√3 x – y = 0 …………. (1)

Slope of the second line = m_{2} = tan (-60°)= – tan 60° = –√3

The equation the second line is y = m_{2} 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 = m_{1} = tan 30° = 1/√3

The equation the first line is y = m_{1} x

y = 1/√3x

√3 y = x

x – √3 y = 0 …………. (1)

Slope of the second line = m_{2} = tan (150°)= tan(180° – 30°) = -tan 30° = -1/√3

The equation the second line is y = m_{2} 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 = m_{1} = tan 30° = 1/√3

The equation the first line is y = m_{1} x

y = 1/√3x

√3 y = x

x – √3 y = 0 …………. (1)

Slope of the second line = m_{2} = tan (150°)= tan(180° – 30°) = -tan 30° = -1/√3

The equation the second line is y = m_{2} 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 = m_{1} = tan 60° = √3

The equation the first line is y = m_{1} x

y = √3x

√3 x – y = 0 …………. (1)

Slope of the second line = m_{2} = tan (120°)= tan (180° – 60°) = – tan 60° = –√3

The equation the second line is y = m_{2} 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 = m_{1} = tan π/3 = √3

The equation the first line is y = m_{1} x

y = √3x

√3 x – y = 0 …………. (1)

Slope of the second line = m_{2} = tan (5π/3)= tan (2π – π/3) = – tan π/3 = –√3

The equation the second line is y = m_{2} 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.

Science > Mathematics > Pair of Straight Lines > You are Here |

Physics |
Chemistry |
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Mathematics |