To express the given parametric equations x = 3t and y = t^2 in polar form, we can substitute these equations into the polar coordinate conversions:
x = r*cos(θ)
y = r*sin(θ)
For convenience, let's solve for r and θ separately:
From x = 3t, we have:
r*cos(θ) = 3t
r = 3t / cos(θ) ----(1)
From y = t^2, we have:
r*sin(θ) = t^2
r = t^2 / sin(θ) ----(2)
Now, equating (1) and (2), we can find the relationship between t, r, and θ:
3t / cos(θ) = t^2 / sin(θ)
Cross-multiplying, we get:
3t * sin(θ) = t^2 * cos(θ)
Dividing both sides by t, we get:
3 * sin(θ) = t * cos(θ)
Now, since t represents the parameter and is typically expressed in terms of θ, we can solve for t in terms of θ:
t = (3 * sin(θ)) / cos(θ)
t = 3 * tan(θ) -----(3)
Finally, we substitute equation (3) back into either equation (1) or (2) to solve for r in terms of θ:
Using equation (1):
r = 3t / cos(θ)
r = 3(3 * tan(θ)) / cos(θ)
r = 9 * tan(θ) / cos(θ)
So, the polar form of the parametric equations x = 3t and y = t^2 is:
r = 9 * tan(θ) / cos(θ)