rewrite each equation in polar form. solve for r if possible
a. 2x+3y=5
b. y=1/2x
2rcosθ + 3rsinθ = 5
r = 5/(2cosθ+3sinθ)
rsinθ = 1/2 rcosθ
tanθ = 1/2
θ = arctan(1/2)
No r involved.
To rewrite the given equations in polar form, we need to express the variables (x and y) in terms of polar coordinates (r and θ).
a. 2x + 3y = 5
To convert x and y to polar coordinates, we use the following equations:
x = r*cos(θ)
y = r*sin(θ)
Substituting these into the given equation, we get:
2(r*cos(θ)) + 3(r*sin(θ)) = 5
Simplifying this equation, we have:
2r*cos(θ) + 3r*sin(θ) = 5
b. y = (1/2)x
Again, we need to express x and y in polar coordinates:
x = r*cos(θ)
y = r*sin(θ)
Substituting these values into the given equation, we have:
r*sin(θ) = (1/2)(r*cos(θ))
This can be further simplified:
r*sin(θ) = (1/2)r*cos(θ)
To solve for r, we can divide both sides of the equation by cos(θ):
r*sin(θ) / cos(θ) = (1/2)r
Simplifying this equation, we obtain:
tan(θ) = 1/2
At this point, we cannot solve for r explicitly because the value of r depends on the specific value of θ chosen. We can find θ using arctan(1/2) and then substitute that value of θ into the equation to find r.