find the eccentricity to identify the conic, then sketch the conic and its directrix
r=1/1-cos theta
r=2/3+2 sin theta
r=4/1-2 cos theta
The general form is
r = k/(1 ± e cosθ)
or
r = k/(1 ± e sinθ)
so, you need to massage what you have into that form
2/(3+2sinθ) = (2/3)/(1 + 2/3 sinθ)
so, e = 2/3
do the others in like wise.
To find the eccentricity of a conic, you can use the equation:
e = √(1 - b²/a²)
where a is the distance between the center and one focus, and b is the distance between the center and one vertex.
Let's solve each equation one by one:
1) Equation: r = 1 / (1 - cos θ)
In this equation, we see that r is defined in terms of θ. This represents a polar equation. To convert it into a Cartesian equation, we can start by using the identity r² = x² + y². Substituting that into the equation:
x² + y² = 1 / (1 - cos θ)
To find the eccentricity, we need to compare this equation to the standard equation for a conic section, which is:
r = ed / (1 - e cos θ)
where e is the eccentricity and d is the distance from the focus to the directrix.
By comparing the two equations, we can see that:
e = 1 / √(1 - cos θ)
Now, let's sketch this conic. Since it is defined in polar coordinates, we will plot points for different values of θ and then convert them into Cartesian coordinates (x, y) using the formulas:
x = r cos θ
y = r sin θ
We can plot points for various values of θ and then connect them to sketch the curve.
2) Equation: r = 2 / (3 + 2 sin θ)
Similar to the previous equation, we need to convert it into Cartesian coordinates before determining the eccentricity and sketching the conic.
Using the same process as before, we can rewrite the equation as:
x² + y² = 2 / (3 + 2 sin θ)
By comparing this equation to the standard conic equation, we can find the eccentricity:
e = 2 / √(3 + 2 sin θ)
Plotting points for different values of θ and converting them into Cartesian coordinates, we can sketch the conic.
3) Equation: r = 4 / (1 - 2 cos θ)
Similarly, we convert this equation into Cartesian coordinates:
x² + y² = 4 / (1 - 2 cos θ)
Comparing it to the standard equation, we find the eccentricity:
e = 2 / √(1 - 2 cos θ)
Plotting points for different values of θ and converting them into Cartesian coordinates, we can sketch the conic.
Once we have the sketch of each conic, we can find the directrix using the formula:
x = d
For each conic, substitute the values of x and y from the equation into the directrix formula to find the equation of the directrix.