Question

Which of the following is not a representation of ? 

	A.
	B.
	C.
	D.

Which of the following is not a representation of ? 

	A.
	B.
	C.
	D.

 

 

Answer 1

(1).

The polar coordinate can be written as (r, θ) = (r, θ + 2nπ) or (r, θ) = [ - r, θ + (2n + 1)π ], where n is any integer.

The polar coordinate point (r, θ₁).

The radius r is same sign then angle θ₁ = θ + 2nπ and solve for n.

The value of n = (θ₁ - θ) / 2π where n must be integer.

The radius r opposite sign then angle θ₁ = θ + (2n + 1)π and solve for n.

The value of n = 1/2 { [ (θ₁ - θ) / π ] - 1 }, where n must be integer.

 

The given polar coordinate point (r, θ₁) = (2, 2π/3).

 

The option A . (2, 7π/3).

Here r = 2 is same sign, so, n = (θ₁ - θ) / 2π.

Substitute θ₁ = 2π/3 and θ = 7π/3 in the above equation.

n = (2π/3 - 7π/3) / 2π

n = (-5π/3) / 2π

n = -5π / 6π = - 5/6 is not an integer, so the point (2, 7π/3) does not represent the point (2, 2π/3).

 

The option B . (2, - 4π/3).

Substitute θ₁ = 2π/3 and θ = - 4π/3 in the above equation.

Here r = 2 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (2π/3 + 4π/3) / 2π

n = (6π/3) / 2π

n = 2π / 2π = 1 is an integer, so the point (2, - 4π/3) represents the point (2, 2π/3).

 

The option C . (-2, -π/3).

Here r = -2 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

Substitute θ₁ = 2π/3 and θ = - π/3 in the above equation.

n = 1/2 { [ (2π/3 + π/3) / π ] - 1 }

= 1/2 { [ (3π/3) / π ] - 1 }

= 1/2 { [3π / 3π ] - 1 }

n = 2π / 2π = 1 is an integer, so the point (2, - π/3) represents the point (-2, 2π/3).

 

The option D . (2, 8π/3).

Here r = 2 is same sign, so,  n = (θ₁ - θ) / 2π.

Substitute θ₁ = 2π/3 and θ = 8π/3 in the above equation.

n = (2π/3 - 8π/3) / 2π

n = (-6π/3) / 2π

n = - 2π / 2π = -1 is an integer, so the point (2, 8π/3) represents the point (2, 2π/3).

 

The option A is correct answer.

 

Answer 2

(2).

The polar coordinate can be written as (r, θ) = (r, θ + 2nπ) or (r, θ) = [ - r, θ + (2n + 1)π ], where n is any integer.

The polar coordinate point (r, θ₁).

The radius r is same sign then angle θ₁ = θ + 360^o and solve for n.

The value of n = (θ₁ - θ) / 360 where n must be integer.

The radius r opposite sign then angle θ₁ = θ + (2n + 1)180^o and solve for n.

The value of n = 1/2 { [ (θ₁ - θ) / 180 ] - 1 }, where n must be integer.

 

The given polar coordinate point (r, θ₁) = (45, 208^o).

 

The option A . (45, -152^o).

Here r = 45 is same sign, so, n = (θ₁ - θ) / 360.

Substitute θ₁ = 208^o and θ = -152^o in the above equation.

n = (208 + 152) / 360

n = 360 / 360

n = 1 is an integer, so the point (45, -152^o) represent the point (45, 208^o).

 

The option B . (-45, -28^o).

Here r = -45 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / 180 ] - 1 }.

Substitute θ₁ = 208^o and θ = -28^o in the above equation.

n = 1/2 { [ (208 + 28) / 180 ] - 1 }

= 1/2 { [ (236) / 180 ] - 1 }

= 1/2 { 56/180 }

n =0.156 is not an integer, so the point (-45, -28^o) does not represent the point (45, 208^o).

 

The option C . (-45, 28^o).

Here r = -45 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / 180 ] - 1 }.

Substitute θ₁ = 208^o and θ = 28^o in the above equation.

n = 1/2 { [ (208 - 28) / 180 ] - 1 }

= 1/2 { [ (180) / 180 ] - 1 }

= 1/2 { 1 - 1 }

n = 0 is an integer, so the point (-45, 28^o) represents the point (45, 208^o).

 

The option D . (-45, -332^o).

Here r = -45 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / 180 ] - 1 }.

Substitute θ₁ = 208^o and θ = -332^o in the above equation.

n = 1/2 { [ (208 + 332) / 180 ] - 1 }

= 1/2 { [ (540) / 180 ] - 1 }

= 1/2 { 3 - 1 }

n = 1 is an integer, so the point (-45, -332^o) represents the point (45, 208^o).

 

The option B is correct answer.