find two sets of parametric equations for the given rectangular equation
x+y^2=4
To find a set of parametric equations for the given rectangular equation x + y^2 = 4, we need to express x and y in terms of another parameter (usually denoted as t).
One way to do this is by assuming y as the parameter t and expressing x in terms of y. We can start by rearranging the equation x + y^2 = 4 to isolate x:
x = 4 - y^2
Now, we can define our first set of parametric equations as follows:
x = 4 - t^2
y = t
In these equations, t can take any real value, and for each value of t, we can substitute it to get corresponding values of x and y.
Another way to find another set of parametric equations is by assuming x as the parameter t and expressing y in terms of x. Let's rearrange the equation x + y^2 = 4 again to isolate y:
y^2 = 4 - x
Taking the square root of both sides, we get:
y = ±√(4 - x)
Now, we can define our second set of parametric equations:
x = t
y = ±√(4 - t)
Again, t can take any real value, and substituting it in the equations will give corresponding values of x and y.
So, the two sets of parametric equations for the given rectangular equation x + y^2 = 4 are:
Set 1:
x = 4 - t^2
y = t
Set 2:
x = t
y = ±√(4 - t)