Question

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.

Answer 1

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) ]