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

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