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Combined Equation of Lines When Lines are Parallel to Axes

Science > Mathematics > Pair of Straight Lines > Joint Equation of Lines When Lines are Parallel to Axes

In this article, we shall study to find a combined or joint equation of pair of lines when lines are parallel to coordinate axes.

Algorithm:
  • When line passes through (h, k) and is parallel to co-ordinate axes are given. Equations of lines are x = h and y = k
  • 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 01:

  • Find the joint equation of a pair of lines which passes through (1, 2) and parallel to co-ordinate axes.
  • Solution:
Joint Equation

Let l1 and l2  be the two lines.

Equation of l1 passing through (1, 2) and parallel to y-axis is x = 1

∴  x – 1 = 0    …………. (1)

Equation of l2 passing through (1, 2) and parallel to  x-axis is y = 2

∴  y – 2 = 0   …………. (2)

The required joint equation is

(x – 1)(y – 2) = 0

∴ xy – 2x – y + 2 = 0

This is the required combined equation.

Example 02:

  • Find the joint equation of a pair of lines which passes through (-3, 4) and parallel to co-ordinate axes.
  • Solution:
Joint Equation

Let l1 and l2  be the two lines.

Equation of l1 passing through (-3, 4) and parallel to y – axis is x = -3

∴  x + 3 = 0    …………. (1)

Equation of l2 passing through (-3, 4) and parallel to  x – axis is y = 4

∴   y – 4 = 0   …………. (2)

The required joint equation is

(x + 3)(y – 4) = 0

∴ xy – 4x + 3y – 12 = 0

This is the required combined equation.

Example 03:

  • Find the joint equation of a pair of lines which passes through (3, 2) and parallel to the lines x = 2 and y = 3.
  • Solution:
Joint Equation

Let l1 and l2  be the two lines.

Equation of l1 passing through (3, 2) and parallel to x = 2 is x = 3

∴  x – 3 = 0    …………. (1)

Equation of l2 passing through (3, 2) and parallel to y = 3 is y = 2

∴   y – 2 = 0   …………. (2)

The required joint equation is

(x – 3)(y – 2) = 0

∴  xy – 2x – 3y + 6 = 0

This is the required combined equation.

Example 04:

  • Find the joint equation of a pair of lines which passes through (2, 3) and parallel to co-ordinate axes.
  • Solution:
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Let l1 and l2  be the two lines.

Equation of l1 passing through (1, 2) and parallel to y – axis is x = 2

∴  x – 2 = 0    …………. (1)

Equation of l2 passing through (1, 2) and parallel to x – axis is y = 3

∴  y – 3 = 0   …………. (2)

The required joint equation is

(x – 2)(y – 3) = 0

∴  xy – 3x – 2y + 6 = 0

This is the required combined equation.

Example 05:

  • Find the joint equation of a pair of lines which are at a distance of 9 units from the y-axis and parallel to it.
  • Solution:
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Let l1 and l2  be the two lines.

Equation of l1 which is at a distance of 9 from y – axis and parallel to it is x = – 9 i.e. x + 9 = 0  …. (1)

Equation of l2 which is at a distance of 9 from y – axis and parallel to it is x =  9 i.e. x – 9 = 0  …… (1)

The required joint equation is

(x + 9)(x – 9) = 0

∴  x2 – 81 = 0

This is the required combined equation.

Example 06:

  • Find the joint equation of a pair of lines which are at a distance of 5 units from the x-axis and parallel to it.
  • Solution:
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Let l1 and l2  be the two lines.

Equation of l1 which is at a distance of 5 from x – axis and parallel to it is y = 5i.e. y – 5  = 0   ……… (1)

Equation of l2 which is at a distance of 5 from x – axis and parallel to it is y =  -5 i.e. y + 5 = 0   …… (2)

The required joint equation is

(y – 5)(y + 5) = 0

∴ y2 – 25 = 0

This is the required combined equation.

Science > Mathematics > Pair of Straight Lines > Combined Equation of Lines When Lines are Parallel to Axes

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