How to find the distance between two points in polar Coordinates?
Let OX be the initial line through the pole O of the polar system and (r₁, θ ₁) and (r₂, θ₂) the polar coordinates 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 xaxis 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 coordinates 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 Coordinates:
Find the length of the linesegment joining the points (4, 10°) and (2√3 ,40°).
Solution:
We know that the length of the linesegment joining the points (r₁, θ₁),and (r₂, θ₂), is
√[ r₂² + r₁²  2r₁ r₂ Cos(θ₂  θ₁)].
Therefore, the length of the linesegment 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.
● Coordinate Geometry
11 and 12 Grade Math
From Distance between Two Points in Polar Coordinates to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.