Chapter 9: Q. 70. (page 732)

Find and prove a formula for the distance between the
points $(r_{1}, θ_{1})$ and $(r_{2}, θ_{1})$ when they are plotted in a polar
coordinate system.

Short Answer

Expert verified

The required answer is

d = \sqrt{r^{2} {}_{1}+ r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{2} - θ_{1})}

Step by step solution

The objective is to find the distance between the points.

Assume that $(r_{1}, θ_{1}) (r_{2}, θ_{2})$ are the polar coordinates of the points $P, Q$

The points are $r_{1} = r_{1} \cos θ_{1}, θ_{1} = r_{1} \sin θ_{1}$ and $r_{2} = r_{2} \cos θ_{2}, θ_{2} = r_{2} \sin θ_{2}$

The distance formula is as follows:

$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

The distance is calculated by substituting the coordinates.

$d = \sqrt{{(r_{2} \cos θ_{2} - r_{1} \cos θ_{1})}^{2} + {(r_{2} \sin θ_{2}, - r_{1} \sin θ_{1})}^{2}} \Rightarrow d = \sqrt{{r_{2}}^{2} \cos^{2} θ_{2} - 2 r_{1} r_{2} \cos θ_{2} + {r_{1}}^{2} \cos^{2} θ_{1} + ({r_{2}}^{2} \sin^{2} θ_{2} - 2 r_{1} r_{2} \sin θ_{1} \sin θ_{2} + {r_{1}}^{2} \sin^{2} θ} \Rightarrow d = \sqrt{{r_{1}}^{2} + {r_{2}}^{2} - 2 r_{1} r_{2} (\sin θ_{1} \sin θ_{2} + \cos θ_{1} \cos_{2})} \Rightarrow d = \sqrt{{r_{1}}^{2} + {r_{2}}^{2} - 2 r_{1} r_{2} (\cos θ_{1} \cos θ_{2} + \sin θ_{1} \sin θ_{2}}$

Find the distance between the points in the polar form

On simplification,

$d = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{2} - θ_{1})} [\sin c e \cos θ_{1} \cos θ_{2} + \sin θ_{1} \sin θ_{2} = \cos (θ_{2} - θ_{1})]$

The distance between the points in the polar form is:

$d = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{2} - θ_{1})}$

Hence the required answer is

$d = \sqrt{r^{2} {}_{1}+ r_{2}^{2} - 2 r_{1} r_{2} \cos (θ_{2} - θ_{1})}$

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Find and prove a formula for the distance between the
points $(r_{1}, θ_{1})$ and $(r_{2}, θ_{1})$ when they are plotted in a polar
coordinate system.

Short Answer

Step by step solution

The objective is to find the distance between the points.

Find the distance between the points in the polar form

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Find and prove a formula for the distance between thepoints (r1,θ1) and (r2,θ1) when they are plotted in a polarcoordinate system.

Short Answer

Step by step solution

The objective is to find the distance between the points.

Find the distance between the points in the polar form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Calculus

Mechanics Maths

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find and prove a formula for the distance between the
points $(r_{1}, θ_{1})$ and $(r_{2}, θ_{1})$ when they are plotted in a polar
coordinate system.