Find a vector parametrization of the ellipse centered at the origin in the xy-plane that has major diameter 8 along the x-axis, minor diameter 4 along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point (4,0) correspond to t=0. Use t as the parameter in your answer.

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jtellezd jtellezd · Answer 1 · 2019-11-10T21:57:43+01:00

Answer:

E(t) = [ 4cos(t) , 2 sin(t) ]

Step-by-step explanation:

The elipse ecuacion is of the form:

(x² ÷ a² ) + (y² ÷ b² ) = 1 Where a and b are the horizontal and vertical semi-diameters respectively

In our particular case x²/16 + y²/ 4 = 1

That form suggests (x/a)² + (y/b)² =1

And again in our case x²/16 + y²/4 = 1 the same shape of sin²α +cos²α = 1

So we call x = 4 cos(t ) and the

We need to find y

x²/16 + y²/4 = 1

x²/16 + y²/4 = 1 ⇒ (x² + 4y²) ÷ 16 = 1

Solving for y (x² + 4y²) = 16 ⇒ y² = ( 16 - x²) ÷4

as x = 4 cos(t) ⇒ y² = (16 - 16cos²(t)) ÷ 4 y = √4 (1 -cos²(t)

y = 2 sin (t)

So the vector parametrization of the elipse is

E(t) = [ 4cos(t) , 2 sin(t) ]