Describe the graph represented by the equation r= 6/3-4cos theta
A: ellipse with a vertical directrix at a distance of 3/2 units to the right of the pole
B: hyperbola with a vertical directrix at a distance of 3 units to the right of the pole
C: hyperbola with a vertical directrix at a distance of 3/2 units to the left of the pole
D: ellipse with a vertical directrix at a distance of 3 units to the left of the pole
D: ellipse with a vertical directrix at a distance of 3 units to the left of the pole.
To describe the graph represented by the equation r = 6/(3 - 4cos(theta)), we can analyze the equation and determine its properties.
First, we can rewrite the equation as r(3 - 4cos(theta)) = 6.
Next, we can distribute r and simplify the equation:
3r - 4rcos(theta) = 6.
Now, we can isolate r by dividing both sides of the equation by 3 - 4cos(theta):
r = 6 / (3 - 4cos(theta)).
This form of the equation suggests that the distance from the origin (pole) to any point on the graph is given by the expression on the right side.
Based on this equation, we can conclude that the graph represented by r = 6/(3 - 4cos(theta)) is an ellipse. An ellipse is a closed curve with two foci, and the distance from any point on the curve to the two foci has a constant sum.
To determine the location of the directrix, we need to consider the coefficient in front of cos(theta). In this case, the coefficient is -4. Since the coefficient is negative, the directrix will be shifted to the left of the pole.
Therefore, the correct answer is option D: an ellipse with a vertical directrix at a distance of 3 units to the left of the pole.
To describe the graph represented by the equation r = 6/(3 - 4cosθ), we need to analyze the equation in polar coordinates.
In polar coordinates, r represents the distance from the origin (pole), and θ represents the angle formed by a line connecting the origin and a point.
Let's simplify the equation first:
r = 6/(3 - 4cosθ)
Multiply both sides of the equation by (3 - 4cosθ):
r(3 - 4cosθ) = 6
Expand the equation:
3r - 4rcosθ = 6
Rearrange the equation:
4rcosθ = 3r - 6
Divide both sides of the equation by 4r:
cosθ = (3r - 6)/(4r)
Now, let's analyze the right side of this equation. The numerator (3r - 6) represents a linear function of r, while the denominator (4r) represents a linear function of r as well.
Cosine (cosθ) takes values between -1 and 1, so for a given angle θ, the value of cosθ must lie between -1 and 1.
If cosθ is between -1 and 1, then we need to check whether the right side of the equation can also lie between -1 and 1. However, since the numerator (3r - 6) is a linear function of r, it can take any value depending on the value of r.
Therefore, it is not possible for the right side of the equation to be equal to cosθ for all values of r.
As a result, there is no well-defined graph for this equation.
Therefore, none of the given options (A, B, C, or D) is correct.