15. Ans: (A)
The general forms of finding all the polar coordinates are:
1) When r >= 0(meaning positive):(r, θ + 2nπ) where, n = integer
2) When r < 0(meaning negative): (-r, θ + (2n+1)π) where, n = integer
Since r = +1, -1(ordered pair)
θ(given) =
When r = +1(r>0):(1, + 2nπ)
When r = -1(r<0):(-1, + (2n+1)π)
Therefore, the correct option is (A) (1, pi divided by 3 + 2nπ) or (-1, pi divided by 3 + (2n + 1)π)
16. Ans: (A)
In polar coordinates,
Since x = 3, y=-3; therefore,
To find the angle,
tanθ = y/x = -3/3 = -1
=> θ = -45°
=> θ = -45°+360° = 315° (when
)
If r = -r =
, then,
θ = -45° + 180° = 135°
Therefore, the correct option is (A) (3 square root of 2 , 315°), (-3 square root of 2 , 135°)
17. Ans: (A)
(Question-17 missing Image is attached below) The general form of the limacon curve is:
r = b + a cosθ
If b < a, the curve would have inner loop.
As you can see in the image attached(labeled Question-17), the limacon curve graph has the inner loop. Therefore, the correct option is (A) r = 2 + 3 cosθ, since b = 2, and a = 3; and the condition b < a (2 < 3) is met.
18. Ans: (B)
Let's find out!
1. If we replace θ with -θ, we would get:
r = 4 - 4*cos(-θ )
Since, cos(-θ) = +cosθ, therefore,
r = 4 - 4*cos(θ)
Same as the original, therefore, graph is symmetric to x-axis.
2. If we replace r with -r, we would get:
-r = 4 - 4*cos(θ )
r = -4 + 4*cos(θ)
NOT same as original, therefore, graph is NOT symmetric to its origin.
3. If we replace θ with -θ and r with -r, we would get:
-r = 4 - 4*cos(-θ )
Since, cos(-θ) = +cosθ, therefore,
r = -4 + 4*cos(θ)
NOT same as original, therefore, graph is NOT symmetric to y-axis.
Ans: The graph is symmetric to: x-axis only!
19. Ans:
Explanation:
As the question suggests that it is a horizontal ellipse, therefore, the equation for the horizontal ellipse is:
-- (A)
Since,
x = 21ft,
y = 29ft,
b = 58ft,
= ?
Plug-in the values in equation (A),
(A)=>
=> = 588
Therefore, the equation becomes,Ans:
20. Ans: x-axis only
Let's find out!
1. If we replace θ with -θ, we would get:
r = 4*cos(-5θ )
Since, cos(-θ) = +cosθ, therefore,
r = +4*cos(5θ) = Same as original
Therefore, graph is symmetric to x-axis.
2. If we replace r with -r, we would get:
-r = 4*cos(5θ )
r = -4*cos(5θ) = Not same
NOT same as original, therefore, graph is NOT symmetric to its origin.
3. If we replace θ with -θ and r with -r, we would get:
-r = 4*cos(-5θ )
Since, cos(-θ) = +cosθ, therefore,
r = -4*cos(5θ) = Not Same
NOT same as original, therefore, graph is NOT symmetric to y-axis.
Ans: The graph is symmetric to: x-axis only!
The general forms of finding all the polar coordinates are:
1) When r >= 0(meaning positive):(r, θ + 2nπ) where, n = integer
2) When r < 0(meaning negative): (-r, θ + (2n+1)π) where, n = integer
Since r = +1, -1(ordered pair)
θ(given) =
When r = +1(r>0):(1,
When r = -1(r<0):(-1,
Therefore, the correct option is (A) (1, pi divided by 3 + 2nπ) or (-1, pi divided by 3 + (2n + 1)π)
16. Ans: (A)
In polar coordinates,
Since x = 3, y=-3; therefore,
To find the angle,
tanθ = y/x = -3/3 = -1
=> θ = -45°
=> θ = -45°+360° = 315° (when
If r = -r =
θ = -45° + 180° = 135°
Therefore, the correct option is (A) (3 square root of 2 , 315°), (-3 square root of 2 , 135°)
17. Ans: (A)
(Question-17 missing Image is attached below) The general form of the limacon curve is:
r = b + a cosθ
If b < a, the curve would have inner loop.
As you can see in the image attached(labeled Question-17), the limacon curve graph has the inner loop. Therefore, the correct option is (A) r = 2 + 3 cosθ, since b = 2, and a = 3; and the condition b < a (2 < 3) is met.
18. Ans: (B)
Let's find out!
1. If we replace θ with -θ, we would get:
r = 4 - 4*cos(-θ )
Since, cos(-θ) = +cosθ, therefore,
r = 4 - 4*cos(θ)
Same as the original, therefore, graph is symmetric to x-axis.
2. If we replace r with -r, we would get:
-r = 4 - 4*cos(θ )
r = -4 + 4*cos(θ)
NOT same as original, therefore, graph is NOT symmetric to its origin.
3. If we replace θ with -θ and r with -r, we would get:
-r = 4 - 4*cos(-θ )
Since, cos(-θ) = +cosθ, therefore,
r = -4 + 4*cos(θ)
NOT same as original, therefore, graph is NOT symmetric to y-axis.
Ans: The graph is symmetric to: x-axis only!
19. Ans:
Explanation:
As the question suggests that it is a horizontal ellipse, therefore, the equation for the horizontal ellipse is:
Since,
x = 21ft,
y = 29ft,
b = 58ft,
Plug-in the values in equation (A),
(A)=>
=>
Therefore, the equation becomes,Ans:
20. Ans: x-axis only
Let's find out!
1. If we replace θ with -θ, we would get:
r = 4*cos(-5θ )
Since, cos(-θ) = +cosθ, therefore,
r = +4*cos(5θ) = Same as original
Therefore, graph is symmetric to x-axis.
2. If we replace r with -r, we would get:
-r = 4*cos(5θ )
r = -4*cos(5θ) = Not same
NOT same as original, therefore, graph is NOT symmetric to its origin.
3. If we replace θ with -θ and r with -r, we would get:
-r = 4*cos(-5θ )
Since, cos(-θ) = +cosθ, therefore,
r = -4*cos(5θ) = Not Same
NOT same as original, therefore, graph is NOT symmetric to y-axis.
Ans: The graph is symmetric to: x-axis only!