Which functions of x and y in terms of time t can be derived from this rectangular equation? (x^2/4)+y^2=1
A.
x = 2sin t, y = -cos t
B.
x = sin t, y = 5cos t
C.
x = sin t, y = 2cos t
D.
x = -sin t, y = 2cos t
E.
x = 2sin t, y = 5cos t
Looks like (A) to me
To determine which functions of x and y in terms of time t can be derived from the given equation (x^2/4) + y^2 = 1, let's analyze the equation and compare it to the options provided.
The given equation is in the form of an ellipse centered at the origin (0, 0), with major axis along the x-axis and minor axis along the y-axis. The general equation of an ellipse is (x^2/a^2) + (y^2/b^2) = 1, where "a" represents the semi-major axis (half the length of the major axis) and "b" represents the semi-minor axis (half the length of the minor axis).
Comparing this general equation to the given equation, we can determine the values of a and b:
(x^2/4) + y^2 = 1
Dividing both sides of the equation by 1, we get:
(x^2/4) + (y^2/1) = 1
Now we can see that the square root of 4 is 2, so the semi-major axis "a" is 2. Similarly, the square root of 1 is 1, so the semi-minor axis "b" is 1.
Knowing the values of a and b, let's analyze the options:
A. x = 2sin t, y = -cos t:
This option represents an ellipse with a = 2 and b = 1. Therefore, it represents a valid solution.
B. x = sin t, y = 5cos t:
This option does not match the given equation because a should be 2, not 1. Therefore, it does not represent a valid solution.
C. x = sin t, y = 2cos t:
This option does not match the given equation because b should be 1, not 2. Therefore, it does not represent a valid solution.
D. x = -sin t, y = 2cos t:
This option represents an ellipse with a = 1, not a = 2. Therefore, it does not represent a valid solution.
E. x = 2sin t, y = 5cos t:
This option represents an ellipse with a = 2 and b = 5. Therefore, it does not represent a valid solution.
Therefore, the only option that represents valid functions of x and y in terms of time t for the given equation is Option A: x = 2sin t, y = -cos t.