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 . m₂ = 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 . m₂ = a/b

∴  km₂ . m₂ = a/b

∴  k(m₂)² = a/b

∴  (m₂)² = a/bk

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 . m₂ = 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 . m₂ = a/b

∴  5m₂ . m₂ = a/b

∴  5(m₂)² = a/b

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 . m₂ = 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 . m₂ = a/b

∴  4m₂ . m₂ = a/b

∴  4(m₂)² = a/b

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 . m₂ = 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 . m₂ = a/b

∴  3m₂ . m₂ = a/b

∴  3(m₂)² = a/b

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 . m₂ = a/b

Given the relation between slopes of lines

m₁ = (m₂)²

Now, m₁ . m₂ = a/b

∴  (m₂)² . m₂ = 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

Multiplying both sides by b³

a²b – 6abh + ab² = -8h³

a²b  + ab² + 8h³ = 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 . m₂ = 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

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.

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 . m₂ = a/b

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