Question

The (planar) curve C parametrized by
r(t) = x(t) i + y(t) j = sin(2t) i + 3 cos(2t) j ; 0 t 3=4
lies on an ellipse in the xy-plane.
(a) Find an equation for this ellipse (in the form
x^2/a^2+y^2/b^2= 1)

Answer 1

well

x(t) = sin(2t)

y(t) = 3cos(2t) for a start...

Answer 2

what does 0 t3=4 mean?

Answer 3

got that part now stuck on the next part
Sketch the curve C, and indicate the direction and the initial point P and nal
point Q (with the coordinates of P and Q).

Answer 4

what does 0 t3=4 mean? ; oh that is a typo what is wanted to write 0 <= t <= 3pi/4

Answer 5

ohh.... 0 to 135 degrees then right?

Answer 6

yeah what did you get for the equation though i got x^2 + y^2/9 = 1

Answer 7

now i don't know about Sketch the curve C, and indicate the direction and the initial point P and nal
point Q (with the coordinates of P and Q).

Answer 8

havent yet :) id have to draw it out; if I see it right; its an ellipse centered at the origin; and then id account for the x and y intercepts to find my vertexes

Answer 9

could you show me how you get to the answer.

Answer 10

sure.... let me write it on paper tho to keep it together

Answer 11

alright

Answer 12

ok:

our vertexes are when x = o and y = 0; which would be at 0,pi/2,pi, and 3pi/2

x = sin(2t) ; y = 3cos(2t)

x = 0 when 2t = 0; y = 3cos(0) = 3(1) = 3

our vertexes at x = 0 are: (0,3) and (0,-3)

y = 0 when 2t = pi/2; x = sin(pi/2) = 1

our vertexes at y=0 are: (1,0) and (-1,0)

you agree?

Answer 13

yes i agree

Answer 14

then yes; the ellipse equation is:

\[\frac{x^2}{1}+\frac{y^2}{9}=1\]

Answer 15

now i don't know about Sketch the curve C, and indicate the direction and the initial point P and nal point Q (with the coordinates of P and Q).

what does this mean?

Answer 16

the curve itself is the ellipse right?

Answer 17

i don't know how to sketch it

Answer 18

let me pull up the paint and see what I get....

Answer 19

this is how it looks on the graph; but I havent checked to see if my domain is accurate

Answer 20

of t is from 0 to 135; that gets us from the (1,0) all the ways to (0,-3) I think

Answer 21

2(135) = 270

Answer 22

and this is what I end up with

Answer 23

does that look right to you?