Question

Which of the following is a representation of ? 

	A.
	B.
	C.
	D.

Which of the following is 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 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) = (75, 50).

 

The option A . (-75, 50).

Here r = -75 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1

Substitute θ₁ = 50 and θ = 50 in the above equation.

n = 1/2 { [ (50-50) / π ] - 1 }

n = (0-1)/2

n = -1/2 is not an integer, so the point (-75,50) does not represent the point (75, 50).

 

The option B . (75, 310).

Substitute θ₁ = 310 and θ = 50 in the above equation.

Here r = 75 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (310-50) / 2π

n = (260 * (π/180)) / 2π

n = 13/18 is not  an integer, so the point (-75,310) does not represent the point (75, 50).

 

The option C . (-75, 230).

Here r = -75 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

Substitute θ₁ = 230 and θ = 50 in the above equation.

n = 1/2 { [ (230-50) / π ] - 1 }

= 1/2 { [ (180 * (π/180)) / π ] - 1 }

= 1/2 { 1 - 1 }

n = 0  is an integer, so the point (-75,230) represent the point (75, 50).

 

The option D . (75, -130).

Here r = 75 is same sign, so,  n = (θ₁ - θ) / 2π.

Substitute θ₁ = -130 and θ = 50 in the above equation.

n = (-130-50) / 2π

n = -π / 2π

n = -1/2 is not an integer, so the point (75,-130) does not represent the point (75, 50).

Therefoer option (c) is correct.

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 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) = (1.8, 70°).

 

The option A . (1.8, 290°).

Here r = 1.8 is same sign, so,  n = (θ₁ - θ) / 2π

Substitute θ₁ = 290 and θ = 70 in the above equation.

n = (290-70) / 2π

n = (220 * (π/180))/2π

n = 11/18 is not an integer, so the point (1.8, 290°) does not represent the point (1.8, 70°).

 

The option B . (1.8, 250°).

Substitute θ₁ = 250 and θ = 70 in the above equation.

Here r = 1.8 is same sign, so,  n = (θ₁ - θ) / 2π.

n = (250-70) / 2π

n = (180 * (π/180)) / 2π

n = 1/2 is not  an integer, so the point (1.8, 250°) does not represent the point (1.8, 70°).

 

The option C . (-1.8,110°).

Here r = -1.8 is opposite sign, so,  n = 1/2 { [ (θ₁ - θ) / π ] - 1 }.

Substitute θ₁ = 110 and θ = 70 in the above equation.

n = 1/2 { [ (110-70) / π ] - 1 }

= 1/2 { [ (40 * (π/180)) / π ] - 1 }

= 1/2 { -7/9 }

n = -7/18  isnot  an integer, so the point (-1.8,110°) represent the point (1.8, 70°).

 

The option D . (1.8, 430°).

Here r = 1.8 is same sign, so,  n = (θ₁ - θ) / 2π.

Substitute θ₁ = -130 and θ = 50 in the above equation.

n = (430-70) / 2π

n = (2π) / 2π

n = 1 is  an integer, so the point (1.8, 430°) does not represent the point (1.8, 70°).

Therefore option (d) is correct.