For the given equation r=1/(3-4sintheta)
a. Graph the equation in polar form.
b. Reduce the equation to simplest rectangular form.
c. Determine the Center and the type of graph of the resulting equation.
r = 1/(3-4sinθ)
(a) see
http://www.wolframalpha.com/input/?i=r+%3D+1%2F%283-4sin%CE%B8%29
(b)
r = 1/(3-4sinθ)
r(3-4sinθ) = 1
3r - 4rsinθ = 1
3√(x^2+y^2) - 4y = 1
3√(x^2+y^2) = 4y+1
3(x^2+y^2) = 16y^2+8y+1
3x^2 - 13y^2 - 8y = 1
3x^2 - 13(y^2 - 8/13 y) = 1
3x^2 - 13(y - 4/13)^2 = 1 + 13(4/13)^2
3x^2 - 13(y - 4/13)^2 = 29/13
39/29 x^2 - 169/29 (y - 4/13)^2 = 1
x^2/(29/39) - (y-4/13)^2/(29/169) = 1
hyperbola with center at (0,4/13)
Steve, your graph does not agree with your solution, so looking at it carefully, I had the same up to
3(√(x^2 + y^2) = 1 + 4y
square both sides
9(x^2 + y^2) = 1 + 8y + 16y^2
9x^2 + 9y^2 - 16y^2 - 8y = 1
9x^2 - 7y^2 - 8y = 1
you forgot to square the 3
I am sure Justine can finish the "completing the square" procedure.
Good catch. I felt vaguely uneasy with the way things worked out, but was in a hurry.
No doubt Justine caught the slip on her own...
a) To graph the equation in polar form, you need to plot several points and connect them to form a curve. Here's how to do it:
1. Choose a range of values for θ, such as -2π to 2π (or 0 to 2π for a complete circle). This will determine the portion of the graph that you want to plot.
2. For each value of θ in that range, calculate the corresponding value of r using the given equation r = 1 / (3 - 4sinθ).
3. Use these values of r and θ to plot the points on a polar coordinate system. The angle θ will determine how far around the circle you go, and r will determine the distance from the origin.
4. Connect the plotted points to form the curve.
b) To reduce the equation to simplest rectangular form, you need to convert the polar equation into a rectangular equation by using the relationships between rectangular (x, y) and polar (r, θ) coordinates. Here's how you can do it:
1. Start with the given polar equation: r = 1 / (3 - 4sinθ).
2. Convert r and θ into rectangular coordinates using the formulas:
x = r * cosθ
y = r * sinθ
3. Substitute these equations into the given polar equation:
x * cosθ + y * sinθ = 1 / (3 - 4sinθ)
4. Simplify the equation by multiplying through by (3 - 4sinθ):
x * cosθ * (3 - 4sinθ) + y * sinθ * (3 - 4sinθ) = 1
5. Expand and simplify:
3x * cosθ - 4x * sinθ * cosθ + 3y * sinθ - 4y * sin²θ = 1
6. Use the trigonometric identity sin²θ + cos²θ = 1 to simplify the equation:
3x * cosθ - 4x * sinθ * cosθ + 3y * sinθ - 4y * (1 - cos²θ) = 1
3x * cosθ - 4x * sinθ * cosθ + 3y * sinθ - 4y + 4y * cos²θ = 1
(3x - 4y) * cosθ - (4x * sinθ - 4y * cosθ) * sinθ = 1 + 4y
This is the simplest rectangular form of the equation.
c) To determine the center and type of graph of the resulting equation, you need to analyze its coefficients. Let's look at the simplified rectangular equation:
(3x - 4y) * cosθ - (4x * sinθ - 4y * cosθ) * sinθ = 1 + 4y
The presence of cosine and sine terms indicates that this equation represents a conic section, specifically an ellipse or a circle. To determine the type of conic section, we need to examine the coefficients:
1. If the coefficients of x² and y² are the same and positive, it represents a circle.
2. If the coefficients of x² and y² are different but have the same sign, it represents an ellipse.
3. If the coefficients of x² and y² have different signs or one of them is zero, it represents a hyperbola or a parabola.
In this case, we have both x and y terms in the equation, so it's not a parabola. Moreover, the coefficients of x² and y² are different and have different signs, so it's not a circle. Therefore, the resulting equation represents an ellipse.
To determine the center of the ellipse, we need to find the values of x and y that make all the coefficients equal to zero. However, since the equation contains trigonometric terms, finding the exact center might be complicated. It's advisable to use graphing software or an online graphing tool to plot the equation and visually determine the center.