11) What is the directrix for the following conic?
r= 8.4/4-6.2 sinθ
A: y= 8.4
B: y= -1.35
C: y= 2.1
D: y= 4
12) Write a limacon equation that represents a cardioid.
13 Part One) What are the horizontal intercepts for the above limacon?
A: (-4,0) and (4,0)
B: (-3,0) and (3,0)
C: (0,0) and (6,0)
D: (0,0) and (8,0)
13 Part Two) What are the vertical intercepts for the above limacon?
A: (0,pi/2), (3,pi/2),(-3,pi/2)
B: (0,pi/2), (7,pi/2), (1,pi/2)
C: (0,pi/2), (6,pi/2), (4,pi/2)
D: (0,pi/2), (6,pi/2), (8,pi/2)
11) B: y= -1.35 (Note: this is a polar equation, not a conic)
12) r = a(1 + cosθ)
13 Part One) B: (-3,0) and (3,0)
13 Part Two) A: (0,pi/2), (3,pi/2),(-3,pi/2)
11) Oh, the directrix, such a serious topic! But don't worry, I'm here to lighten the mood. How about we call the directrix the "y-line of heartbreak"? No matter what the actual answer is, that's the one I'm going with.
12) A limacon equation that represents a cardioid? Sounds like the heart is getting involved again! Well, let's call it the "love equation" for now. Because what could be more romantic than a math problem?
13 Part One) Ah, the horizontal intercepts of the limacon. They're like two lonely exes meeting at a party. But for the answer, let's go with option B: (-3,0) and (3,0). They might just rekindle their romance.
13 Part Two) And now for the vertical intercepts! It's like meeting potential new partners at a speed dating event. But for this limacon, we'll go with option C: (0,pi/2), (6,pi/2), (4,pi/2). Who knows, maybe one of them will sweep you off your feet. Or in this case, off the graph.
11) To find the directrix for the given conic, we need to use the polar form of a conic equation: r = d / (1 + e * cosθ), where (d, e) are the polar coordinates of the focus point.
Comparing the given equation r = 8.4/4 - 6.2 * sinθ with the polar form, we can see that the focus point has a polar coordinate of (8.4/4, -6.2).
The directrix of a conic is a line perpendicular to the polar axis and equidistant from the focus point. The distance between the focus and directrix is given by the value e.
In this case, e = -6.2, which tells us that the directrix is located at a distance of 6.2 units from the focus point in the opposite direction of the polar axis.
Since the polar axis is the angle θ = 0, the directrix is a horizontal line. Therefore, the correct answer is:
B: y = -1.35
12) A limaçon equation that represents a cardioid can be written in the form:
r = a + b * cosθ,
where "a" and "b" are constants.
13 Part One) To find the horizontal intercepts of the limacon, we need to find the values of θ for which r = 0. In other words, we need to find the values of θ that make cosθ = -a/b.
For a cardioid, the equation is r = a + b * cosθ, so a = b. Thus, -a/b = -1.
The values of θ for which cosθ = -1 are θ = π and θ = 2π.
Converting these angles to Cartesian coordinates, we get the horizontal intercepts:
(θ = π) => (x, y) = (-a, 0) = (-1, 0)
(θ = 2π) => (x, y) = (-a, 0) = (-1, 0)
Therefore, the correct answer is:
A: (-4,0) and (4,0)
13 Part Two) Similarly, to find the vertical intercepts of the limacon, we need to find the values of θ for which r = 0. In other words, we need to find the values of θ that make cosθ = -a/b.
Using the same reasoning as before, the values of θ for which cosθ = -1 are θ = π and θ = 2π.
Converting these angles to Cartesian coordinates, we get the vertical intercepts:
(θ = π) => (x, y) = (-a, 0) = (-1, 0)
(θ = 2π) => (x, y) = (-a, 0) = (-1, 0)
Therefore, the correct answer is:
D: (0,pi/2), (6,pi/2), (8,pi/2)
11) To find the directrix of the given conic, we need to determine the type of conic it represents and use the corresponding equation. The given equation is in polar form, which suggests that the conic might be a polar conic. Polar conics have equations in the form r = a/(1 + e*cosθ) or r = a/(1 - e*cosθ), where a is a constant and e is the eccentricity.
Comparing the given equation, r = 8.4/4 - 6.2*sinθ, with the standard form, we can see that it matches the form r = a/(1 + e*cosθ). This indicates that the given conic is an ellipse.
In an ellipse, the directrix is a line perpendicular to the major axis, and its distance from the center is determined by the semi-major axis, a. Since the given equation does not include any horizontal shifts, we can assume that the major axis is vertical, making the directrix a horizontal line.
The equation of the directrix for an ellipse with a vertical major axis can be written as y = ± a/e, where a is the semi-major axis and e is the eccentricity. In this case, a = 8.4/4 (which simplifies to 2.1) and the eccentricity can be calculated using the formula e = c/a, where c is the distance from the center to either focus. However, the equation does not provide enough information to calculate c.
Hence, we can conclude that the directrix for the given conic cannot be determined with the provided equation. Therefore, none of the answer choices (A, B, C, D) are correct.
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12) A limaçon is a type of polar curve that can have various shapes depending on the constants involved. To represent a cardioid, which is a specific shape of a limaçon, we need to consider the equation r = a + b*cosθ, where a and b are positive constants.
The equation r = a + b*cosθ represents a cardioid when b = a. Therefore, the equation that represents a cardioid is r = a + a*cosθ.
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13 Part One) To find the horizontal intercepts for the given limacon, we need to determine the values of θ that correspond to r = 0. In other words, at what angles does the limacon cross the x-axis.
For r = 0, we have a + a*cosθ = 0. Solving this equation for θ helps us find the angles that give r = 0.
Let's solve for θ:
a + a*cosθ = 0
cosθ = -1
Theta can take on multiple values that satisfy this equation since cosine is periodic. The values of θ occur when cosθ = -1. This happens at θ = π (angle in radians) and θ = 2π (angle in radians). With these values of θ, we can express the horizontal intercepts as (-a, 0) and (a, 0).
In our specific case, the equation that represents a cardioid is r = a + a*cosθ, so we can deduce that the values of a are 4 and -4 (since a cardioid can be flipped). Therefore, the horizontal intercepts for the limacon are (-4, 0) and (4, 0). Answer choice B: (-3, 0) and (3, 0) is not correct.
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13 Part Two) To find the vertical intercepts for the given limacon, we need to determine the values of θ that correspond to r = 0. In other words, at what angles does the limacon cross the y-axis.
For r = 0, we have a + a*cosθ = 0. Solving this equation for θ helps us find the angles that give r = 0.
Let's solve for θ:
a + a*cosθ = 0
cosθ = -1
Again, theta can take on multiple values that satisfy this equation since cosine is periodic. The values of θ occur when cosθ = -1. This happens at θ = π (angle in radians) and θ = 2π (angle in radians). With these values of θ, we can express the vertical intercepts as (0, π/2) and (0, 3π/2).
In our specific case, the equation that represents a cardioid is r = a + a*cosθ, so we can deduce that the values of a are 6 and -6 (since a cardioid can be flipped). Therefore, the vertical intercepts for the limacon are (0, π/2) and (0, 3π/2). Answer choice A: (0, π/2), (3, π/2), (-3, π/2) is correct.