Convert the following equation from an equation using polar coordinates to an equation using rectangular coordinates: 6sinθ=r-5cosθ
a.) 6y=-5x
b.) x^2-6x+y^2-5y=0
c.) x^2-5x+y^2-6y=0
d.) x^2+6x+y^2+5y=0
e.) x^2+5x+y^2+6y=0
f.) 5y=-6x
6sinθ=r-5cosθ
6rsinθ = r^2 - 5rcosθ
6y = x^2+y^2 - 5x
now just rearrange the terms
To convert the equation from polar coordinates to rectangular coordinates, we can use the following relationships:
x = rcosθ
y = rsinθ
Let's use these relationships to rewrite the equation:
6sinθ = r - 5cosθ
Rewriting in rectangular coordinates:
6y = x - 5(-y)
Simplifying:
6y = x + 5y
Combining like terms:
6y - 5y = x
y = x
Therefore, the correct answer is:
a.) 6y = -5x
To convert an equation from polar coordinates to rectangular coordinates, we use the following conversions:
x = r * cos(θ)
y = r * sin(θ)
Let's apply these conversions to the given equation:
6sinθ = r - 5cosθ
Now, substitute x and y in terms of r and θ:
6y = r - 5x
6(r * sin(θ)) = r - 5(r * cos(θ))
6r * sin(θ) = r - 5r * cos(θ)
Now, divide through by r:
6 * sin(θ) = 1 - 5 * cos(θ)
We now have the equation in terms of r and θ, but to convert it to rectangular coordinates, we need to further simplify it.
To eliminate θ, we use the fact that sin²(θ) + cos²(θ) = 1. Rearranging, we have sin(θ) = √(1 - cos²(θ)). Substitute this into the equation:
6 * √(1 - cos²(θ)) = 1 - 5 * cos(θ)
Solving for cos(θ):
36 * (1 - cos²(θ)) = (1 - 5 * cos(θ))²
36 - 36cos²(θ) = 1 - 10cos(θ) + 25cos²(θ)
Now, combine like terms:
-61cos²(θ) + 10cos(θ) - 35 = 0
This is a quadratic equation in terms of cos(θ). To solve for cos(θ), we can use the quadratic formula:
cos(θ) = (-b ± √(b² - 4ac)) / (2a)
where a = -61, b = 10, and c = -35.
After determining the values of cos(θ), substitute them back into the equation 6 * √(1 - cos²(θ)) = 1 - 5 * cos(θ) to find the values of sin(θ). Finally, substitute the values of cos(θ) and sin(θ) back into the rectangular coordinate equations x = r * cos(θ) and y = r * sin(θ) to obtain the equation in rectangular coordinates. Once the equation is simplified, you can identify the correct answer among the options provided.