Question

Describe the motion of a particle with position (x, y) as t varies in the given interval. (For each answer, enter an ordered pair of the form x, y.) x = 1 + sin(t), y = 6 + 2 cos(t), π/2 ≤ t ≤ 2π The motion of the particle takes place on an ellipse centered at (x, y) = 1,6 . As t goes from π/2 to 2π, the particle starts at the point (x, y) = 1,6 and moves clockwise three-fourths of the way around the ellipse to (x, y) = .

Answer 1

Two foci of the ellipse are ( 2 - √3) and moves clockwise three fourth is ( 2 + √3).

As given in the question,

Given equations are :

x = 1 + sin(t)

⇒ sin(t) = (x - 1)

y = 6 + 2cos(t)

⇒cos(t) = (y - 6)/2

We know that,

sin²t + cos²t = 1

Substitute the value of sin(t) and cos(t) we get,

(x - 1)²/ 1² + (y - 6)²/ 2² = 1

Standard Equation of the ellipse  is:

( x -h)²/b² + (y - k)²/a² = 1

Center of the ellipse = (h , k)

(h , k) = (1, 6)

here b = 1 and a = 2

'c' is the distance from focus to the center

c = √a² - b²

  = √4 -1

  = √3

Therefore , two foci for the given ellipse are ( 2 - √3) and moves clockwise three fourth is ( 2 + √3).

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