Find the rectangular equation for the given polar coordinates:
R=(7/(5+4sin theta))
r = 7/(5+4sinθ)
r(5+4sinθ) = 7
5r + 4rsinθ = 7
5r = 7 - 4y
25r^2 = (7-4y)^2
25x^2 + 25y^2 = 49 - 56y + 16y^2
25x^2 + 9y^2 + 56y = 49
25x^2 + 9(y+28/9)^2 = 49 + 9(28/9)^2
25x^2 + 9(y+28/9)^2 = 1225/9
x^2/9 + (y+28/9)^2/25 = (7/9)^2
See
http://www.wolframalpha.com/input/?i=r+%3D+7%2F%285%2B4sin%CE%B8%29
To find the rectangular equation for the given polar coordinates (R, θ), where R = 7 / (5 + 4sin θ), we can use the following steps:
Step 1: Recall the relationship between polar and rectangular coordinates.
In polar coordinates, a point (R, θ) represents a point in the Cartesian coordinate system, where R is the distance from the origin (0,0) to the point and θ is the angle formed by the horizontal axis and the line connecting the origin to the point.
Step 2: Express R in terms of x and y.
The distance from the origin to a point (x, y) in the Cartesian coordinate system can be calculated using the Pythagorean theorem:
R = √(x^2 + y^2)
Step 3: Express sin θ in terms of x and R.
Using the definition of sine, we can relate sin θ to the ratio of the opposite side to the hypotenuse in a right triangle. In this case, the opposite side is y, and the hypotenuse is R:
sin θ = y / R
Step 4: Substitute the expression for sin θ into the equation for R.
R = 7 / (5 + 4sin θ)
R = 7 / (5 + 4(y / R))
R^2 = 7R / (5R + 4y)
R(5R + 4y) = 7R^2
Expand the equation:
5R^2 + 4yR = 7R^2
Step 5: Rearrange the equation to obtain the rectangular equation.
Move all the terms to one side:
7R^2 - 5R^2 - 4yR = 0
Simplify:
2R^2 - 4yR = 0
Factor out R:
R(2R - 4y) = 0
The rectangular equation for the given polar coordinates R = 7 / (5 + 4sin θ) is 2R^2 - 4yR = 0.