find two sets of parametric equations for the given rectangular equation.
x^2+4y^2-16=0
x^2 - 16 = 4y^2 or (x+4)(x-4) = 4y^2
let x = t
then
t^2 - 16 = 4y^2
y^2 = (t^2 - 16)/4
y = ± √(t^2 - 16)/2 , -4 ≤ t ≤ 4
x = 4cos(t)
y = 2sin(t)
x^2+4y^2 = 16cos^2(t)+16sin^2(t) = 16
To find the parametric equations for the given rectangular equation x^2 + 4y^2 - 16 = 0, we can use the following steps:
Step 1: Solve the equation for x or y.
Step 2: Express x or y in terms of a parameter t.
Step 3: Choose a suitable range for the parameter t.
Set 1: Parametric equations with x as the parameter:
Step 1: Solve for x.
x^2 + 4y^2 - 16 = 0
x^2 = 16 - 4y^2
x = ±√(16 - 4y^2)
Step 2: Express x in terms of the parameter t.
Let x = √(16 - 4y^2)
Step 3: Choose a suitable range for the parameter t.
Since x = √(16 - 4y^2), we can choose the parameter t to represent y in the range [-2, 2].
Therefore, the parametric equations for Set 1 are:
x = √(16 - 4y^2)
y = t, where t is in the range [-2, 2]
Set 2: Parametric equations with y as the parameter:
Step 1: Solve for y.
x^2 + 4y^2 - 16 = 0
4y^2 = 16 - x^2
y^2 = (16 - x^2) / 4
y = ±√((16 - x^2) / 4)
Step 2: Express y in terms of the parameter t.
Let y = √((16 - x^2) / 4)
Step 3: Choose a suitable range for the parameter t.
Since y = √((16 - x^2) / 4), we can choose the parameter t to represent x in the range [-4, 4].
Therefore, the parametric equations for Set 2 are:
x = t, where t is in the range [-4, 4]
y = √((16 - x^2) / 4)
To find the parametric equations for the given rectangular equation, we can use the following approach:
Step 1: Solve the equation for one of the variables in terms of the other variable.
Step 2: Express both variables in terms of a parameter.
Step 3: Write the parametric equations using the parameter and the expressions obtained in Step 2.
Let's go through these steps to find two sets of parametric equations for the given rectangular equation x^2 + 4y^2 - 16 = 0:
Step 1: Solve for x in terms of y.
x^2 + 4y^2 - 16 = 0
x^2 = 16 - 4y^2
x = ±√(16 - 4y^2)
Step 2: Express both x and y in terms of a parameter, say t.
Let's choose t = y as the parameter.
Step 3: Write the parametric equations.
Using t = y as the parameter, the two sets of parametric equations for the given rectangular equation are:
First set:
x = √(16 - 4t^2)
y = t
Second set:
x = -√(16 - 4t^2)
y = t
These parametric equations represent two sets of curves that satisfy the given rectangular equation. By substituting different values of the parameter t, you can generate various points on these curves.