Find a parametrization of the ellipse centered at the origin in the xy-plane tha

Question

Find a parametrization of the ellipse centered at the origin in the xy-plane that has major diameter along the x-axis, minor diameter along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point correspond to . Use as the parameter for all of your answers.

Find a parametrization of the ellipse centered at the origin in the xy-plane that has major diameter 14 along the x-axis, minor diameter 10 along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point (7, 0)correspond to t = 0. Use t as the parameter for all of your answers. x(t) = and y(t) =

Explanation / Answer

The equation of an ellipse with center (0,0) is:
x^2/a^2 + y^2/b^2 = 1

a and b represent the major and minor axes, which are similar to the radius of a circle and are equal to half the major and minor diameters, respectively.

For an ellipse with major diamter 14 along the x-axis and minor diamter 10 along the y-axis:

14 = 2a

10 = 2b

a = 7, b = 5

The parametrization of an ellipse is analogous to that of a circle but with x and y scaled by the:
x = a cos t
y = b sin t

r = xi + yj

r = (7 cos t)i + (5 sin t)j

At t = 0, x = 7 and y = 0, which corresponds to the point (7,0)

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