Questions and detailed answers on Parametric equations and polar coordinates

Here are sample questions with
solutions to help you understand
parametric equations and polar
coordinates.

Sample Question 1:
Find the Cartesian equation of
the curve given by x = 3t^2 + 2t,
y = 4t^2 + t.

Solution 1:
To obtain the Cartesian equation,
we need to eliminate the
parameter t. From the given
parametric equations, we have:

t = (y - t^2)/4 = (x - 3t^2)/2

Simplifying this, we get:

,4t = y - t^2
2t = x - 3t^2

Substituting for t in the second
equation,

,we get:

2(y - t^2)/4 = x - 3(y - t^2)^2/36

Simplifying this, we get:

9x - 4y = 5y^2 - 14ty + 3t^2

Substituting for t in terms of x
and y, we get the Cartesian
equation:

9x - 4y = 5y^2 - 14y(x - y^2/4) +
3(x - y^2/4)^2

Simplifying this, we get:

x^2 + y^2 - 4x + 3y^2 = 0

, This is the Cartesian equation of
the curve.

Sample Question 2: