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To Find Condition When Relation Between Slopes f lines is Given

Science > Mathematics > Pair of Straight Lines > To Find the Condition When Relation Between Slopes is Given

In this article, we shall discuss problems to find the condition when the relation between slopes of lines represented by a joint equation is given.

Example – 18:

  • If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is k times the slope of the other, prove that 4kh2 = ab(1+ k)2.
  • Solution:

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

Given the relation between slopes of lines

m1 = km2

Now, m1 + m2 = – 2h/b

∴  km2 + m2 =- 2h/b

∴ (k + 1)m2 =- 2h/b

∴  m2 = – 2h/b(k + 1)

Now, m1 . m= a/b

∴  km2 . m= a/b

∴  k(m2)2 = a/b

∴  (m2)2 = a/bk

relation between slopes of lines

4kh2 = ab(1+ k)2 (proved as required)

Example – 19:

  • If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is 5 times the slope of the other, prove that 5h2 = 9ab.
  • Solution:

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

Given the relation between slopes of lines

m1 = 5m2

Now, m1 + m2 = – 2h/b

∴  5m2 + m2 =- 2h/b

∴    6m2 =- 2h/b

∴  m2 = – h/3b

Now, m1 . m= a/b

∴  5m2 . m= a/b

∴  5(m2)2 = a/b

relation between slopes of lines

5h2 = 9ab  (proved as required)

Example – 20:

  • If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is 4 times the slope of the other, prove that 16h2 = 25ab.
  • Solution:

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

Given the relation between slopes of lines

m1 = 4m2

Now, m1 + m2 = – 2h/b

∴  4m2 + m2 =- 2h/b

∴    5m2 =- 2h/b

∴  m2 = – 2h/5b

Now, m1 . m= a/b

∴  4m2 . m= a/b

∴  4(m2)2 = a/b

relation between slopes of lines

16h2 = 25ab  (proved as required)

Example – 21:

  • If the slope of one of the lines given by ax2 + 2hxy + by2 = 0 is 3 times the slope of the other, prove that 3h2 = 4ab.
  • Solution:

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

Given the relation between slopes of lines

m1 = 3m2

Now, m1 + m2 = – 2h/b

∴  3m2 + m2 =- 2h/b

∴    4m2 =- 2h/b

∴  m2 = – h/2b

Now, m1 . m= a/b

∴  3m2 . m= a/b

∴  3(m2)2 = a/b

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3h2 = 4ab  (proved as required)

Example – 22:

  • Find the condition that slope of one of the lines represented by ax2 + 2hxy  + by2 = 0 is square of the slope of another line.
  • Solution:

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

Given the relation between slopes of lines

m1 = (m2)2

Now, m1 . m= a/b

∴  (m2)2 . m= a/b

∴  (m2)3 = a/b

Now, m1 + m2 = – 2h/b

∴  (m2)2 + m2 = – 2h/b

Making cube of both sides

((m2)2 + m2)3 = (- 2h/b)3

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Multiplying both sides by b3

a2b – 6abh + ab2 = -8h3

a2b  + ab2 + 8h= 6abh

This is the required condition

Example – 23:

  • Show that one of the straight line  given by ax2 + 2hxy + by2 = 0 bisects an angle between the co-ordinate axes if and only if  (a + b)2 = 4h2.
  • Solution:

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

Now given that One of the straight line represented by the joint equation bisects the angle between the co-ordinate axes. Thus the slope of the line is ± 1. let m2 = ± 1

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Example – 24:

  • If the straight lines represented by a joint equation  ax2 + 2hxy + by2 = 0 make angles of equal measures with the co-ordinate axes. Then prove that a = ± b.
  • Solution:

Let α be the angle made by the lines with coordinate axes.

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Let OA be the line making an angle α with the x-axis and

OB be the line making an angle α with y-axis.

The angle made by the line with the positive direction of x-axis is α. Hence its slope is

m1 = tan α

The angle made by OB with the positive direction of x-axis is (π/2 ± α). Hence its slope is

m2 = tan (π/2 ± α) = ± cot α

The given joint equation is ax2 + 2hxy + by2 = 0

Let m1 and m2 be the slopes of the two lines represented by given joint equation

 (m1 + m2) = – 2h/b, and  m1 . m= a/b

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Science > Mathematics > Pair of Straight Lines > To Find the Condition When Relation Between Slopes is Given

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