Eliminate the parameter and write the corresponding rectangular equation. X=2cos theta, y=4sin theta
To eliminate the parameter and write the corresponding rectangular equation, we need to express theta in terms of x and y.
First, let's consider the equation x = 2cos(theta). To get rid of the parameter, we can solve this equation for cos(theta) in terms of x:
x = 2cos(theta)
cos(theta) = x/2
Next, let's look at the equation y = 4sin(theta). Similarly, we can solve this equation for sin(theta) in terms of y:
y = 4sin(theta)
sin(theta) = y/4
Now, we have expressions for cos(theta) and sin(theta) in terms of x and y. To eliminate the parameter, we can use the trigonometric identity:
cos^2(theta) + sin^2(theta) = 1
Substituting the expressions for cos(theta) and sin(theta), we get:
(x/2)^2 + (y/4)^2 = 1
Expanding and rearranging the equation, we have:
x^2/4 + y^2/16 = 1
This is the corresponding rectangular equation that eliminates the parameter.
To eliminate the parameter, we need to use trigonometric identities to express x and y in terms of each other without the parameter theta.
Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can solve for cos^2(theta):
cos^2(theta) = 1 - sin^2(theta)
Now, let's substitute X = 2cos(theta) and Y = 4sin(theta) into the equation:
X^2 = (2cos(theta))^2 = 4cos^2(theta)
Y^2 = (4sin(theta))^2 = 16sin^2(theta)
Since cos^2(theta) = 1 - sin^2(theta), we can rewrite the first equation as:
X^2 = 4(1 - sin^2(theta)) = 4 - 4sin^2(theta)
Hence, the corresponding rectangular equation is:
X^2 + 4Y^2 = 4 - 4sin^2(theta) + 16sin^2(theta)
==> X^2 + 4Y^2 = 4 + 12sin^2(theta)
x/2 = cos theta and y/4 = sin theta
but sin^2 (theta) + cos^2 (theta) = 1
x^2/4 + y^2/16 = 1