Distance between Two Points in Polar Co-ordinates

How to find the distance between two points in polar Co-ordinates?

Let OX be the initial line through the pole O of the polar system and (r₁, θ ₁) and (r₂, θ₂) the polar co-ordinates of the points P and Q respectively. Then, OP₁ = r₁, OQ = r₂, ∠XOP = θ₁ and ∠XOQ = θ₂, Therefore, ∠POQ = θ₂ – θ₁. 

From triangle POQ we get,

PQ² = OP² + OQ² – 2 ∙ OP ∙ OQ ∙ cos∠POQ

    = r₁² + r₂² – 2r₁ r₂ cos(θ₂ - θ₁)

Therefore, PQ = √[r₁² + r₂ ² - 2r₁ r₂ cos⁡(θ₂ - θ₁)].

Second Method: Let us choose origin and positive x-axis of the cartesian system as the pole and initial line respectively of the polar system. If (x₁, y₁) , (x₂, y₂) and (r₁, θ₁) (r₂, θ₂) be the respective Cartesian and polar co-ordinates of the points P and Q, then we shall have,

    x₁ = y₁ cos θ₁,     y₁ = r₁ sin θ₁

and

    x₂ = r₂ cos θ₂,     y₂ = r₂ sin θ₂.

Now, the distance between the points P and Q is

PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]

     = √[(r₂ cos θ₂ - r₁ cos θ₁)² + (r₂ sin θ₂ - r₂ sin θ₂)²]

     = √[r₂² cos² θ₂ + r₁ ² cos² θ₁ - 2 r₁r₂ cos θ₁ cos θ₂ + r₂² sin² θ₂ + r₁²sin² θ₁ - 2 r₁r₁ sin θ₁ sin θ₂]

     = √[r₂² + r₁² - 2r₁ r₂ Cos(θ₂ - θ₁)].

Example on distance between two points in polar Co-ordinates:

Find the length of the line-segment joining the points (4, 10°) and (2√3 ,40°).

Solution:

We know that the length of the line-segment joining the points (r₁, θ₁),and (r₂, θ₂), is

     √[ r₂² + r₁² - 2r₁ r₂ Cos(θ₂ - θ₁)].

Therefore, the length of the line-segment joining the given points

     = √{(4² + (2√3)² - 2 ∙ 4 ∙ 2√(3) Cos(40 ° - 10°)}

     = √(16 + 12 - 16√3 ∙ √3/2)

     = √(28 - 24)

     = √4

     = 2 units.

● Co-ordinate Geometry 

• What is Co-ordinate Geometry?
  
• Rectangular Cartesian Co-ordinates
  
• Polar Co-ordinates
  
• Relation between Cartesian and Polar Co-Ordinates
  
• Distance between Two given Points
  
• Distance between Two Points in Polar Co-ordinates
  
• Division of Line Segment: Internal & External
  
• Area of the Triangle Formed by Three co-ordinate Points
  
• Condition of Collinearity of Three Points
  
• Medians of a Triangle are Concurrent
  
• Apollonius' Theorem
  
• Quadrilateral form a Parallelogram 
  
• Problems on Distance Between Two Points 
  
• Area of a Triangle Given 3 Points
  
• Worksheet on Quadrants
  
• Worksheet on Rectangular – Polar Conversion
  
• Worksheet on Line-Segment Joining the Points
  
• Worksheet on Distance Between Two Points
  
• Worksheet on Distance Between the Polar Co-ordinates
  
• Worksheet on Finding Mid-Point
  
• Worksheet on Division of Line-Segment
  
• Worksheet on Centroid of a Triangle
  
• Worksheet on Area of Co-ordinate Triangle
  
• Worksheet on Collinear Triangle
  
• Worksheet on Area of Polygon
  
• Worksheet on Cartesian Triangle

11 and 12 Grade Math 

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