6) rewrite (x+3)^2+y^2=9 in polar form.
A: r^2= 3rsin theta
B: r^2= 9rcos theta
C: r= -6 cos theta
D: r= 3rsin theta
7)Which of the following types of symmetry does the graph of the equation r-8 sin 7 theta have?
A: Symmetry about the horizontal axis
B: Symmetry about the vertical axis
C: Symmetry about the pole
D: No symmetry
8) The graph of r=4 sin theta has which of the following characteristics?
A: circle, diameter of 4, center at (0,2)
B: circle, diameter of 4, center at (2,0)
C: circle, radius of 2 center (4,0)
D: circle, radius of 2, center at (0,4)
D: circle, radius of 2, center at (0,4)
AAAaannndd the bot gets it wrong yet again!
6) (x+3)^2+y^2=9
Clearly, the center is at (-3,0) and the radius is 3.
That makes the polar form r = -6cosθ
7) assuming a typo, and you meant r = 8 sin7θ
then it is a rosette with an odd number of petals, and its axis of symmetry is the vertical axis (since r reaches a maximum at θ = 3π/2)
8) r=4 sinθ
has a radius of 2, with center at (0,2)
because we have
r^2 = 4rsinθ
x^2+y^2 = 4y
x^2 + (y-2)^2 = 4
My apologies for the mistake in my previous response. Thank you for correcting me.
6) r = -6cosθ
7) Assuming the equation is actually r = 8sin(7θ), then it has symmetry about the polar axis (θ = 0) and also symmetry about the lines θ = π/7 and θ = 6π/7.
8) Circle, radius of 2, center at (0, 2).
6) To rewrite the equation (x+3)^2 + y^2 = 9 in polar form, we need to replace x and y with their corresponding polar representations.
Using the conversion formulas x = r cos(theta) and y = r sin(theta), we can rewrite the equation as:
(r cos(theta) + 3)^2 + (r sin(theta))^2 = 9
Expanding and simplifying, we get:
r^2 cos^2(theta) + 6r cos(theta) + 9 + r^2 sin^2(theta) = 9
Since cos^2(theta) + sin^2(theta) = 1, we can simplify further to:
r^2 (cos^2(theta) + sin^2(theta)) + 6r cos(theta) = 0
Simplifying again, we obtain:
r^2 + 6r cos(theta) = 0
Dividing both sides by r, we get:
r + 6 cos(theta) = 0
Therefore, the equation in polar form is:
r = -6 cos(theta)
So, the correct answer is: C) r = -6 cos(theta)
7) To determine the type of symmetry the graph of the equation r - 8 sin(7 theta) has, we need to analyze its components.
The term r represents the distance from the origin, while 8 sin(7 theta) represents the vertical component.
Since the equation only involves the vertical component, we can conclude that the graph will have symmetry about the vertical axis. This means that if we reflect the graph across the vertical axis, it will look the same.
Therefore, the correct answer is: B) Symmetry about the vertical axis
8) The equation r = 4 sin(theta) represents a graph in polar coordinates.
In polar coordinates, r represents the distance from the origin, and theta represents the angle with the positive x-axis.
We can see that the equation only involves the sine function, which creates a graph that oscillates between positive and negative values.
Since the coefficient of sin(theta) is 4, it determines the maximum value of r. In this case, the maximum value is 4.
Additionally, the graph of a circle in polar coordinates has the form r = a, where a is the radius of the circle. In this equation, r = 4 sin(theta), we can identify that the graph is indeed a circle.
So, the correct answer is: C) circle, radius of 2, center at (4,0)