Find a parametric equation for the ellipse. Make sure that the graph of your parametric equation is a complete ellipse.
3x^2+4y^2=12
3x^2 + 4y^2 = 12
divide by 12
x^2/4 + y^2/3 = 1
we know sin^2 t + cos^2 t = 1
so sin^2 t = x^2/4 = (x/2)^2
---> sin t x/s ----> x = 2sin t
and
cos^2 t = y^2/3 = (y/√3)^2
cos t = y/√3
----> y = √3cos t
x = 2sint
y = √4cost
confirmation:
http://www.wolframalpha.com/input/?i=x+%3D+2sint+%2C+y+%3D+√3cost
To find a parametric equation for the ellipse 3x^2 + 4y^2 = 12, we can use the following steps:
Step 1: Rewrite the equation in standard form:
Divide both sides of the equation by 12 to obtain:
x^2/4 + y^2/3 = 1
Step 2: Identify the values of a and b:
From the standard form equation, we can see that a^2 = 4 and b^2 = 3. Therefore, a = 2 and b = √3.
Step 3: Define the parameter:
Let t be a parameter that ranges from 0 to 2π.
Step 4: Define x and y in terms of t:
x = a * cos(t)
Since a = 2, the equation becomes:
x = 2 * cos(t)
y = b * sin(t)
Since b = √3, the equation becomes:
y = √3 * sin(t)
Step 5: Combine the equations to form the parametric equations:
The parametric equations for the ellipse 3x^2 + 4y^2 = 12 are:
x = 2 * cos(t)
y = √3 * sin(t)
These parametric equations describe a complete ellipse because as t ranges from 0 to 2π, both x and y will trace out the entire ellipse, ensuring that the graph is complete.