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
11 and 12 Grade Math
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