parametric equations

用戶110869

2007-05-01 11:44 pm

The parametric equations of the locus of a movable point P is given by
x = a^2 cos t
y = b^2 sin t

(a) Find the time taken of the first time that P reaches the original position.
(b) Find the speed of P when it reaches the original position.

更新1:

(c) There are two points A and B such that the sum of the length of AP and BP is always the same. ˉ(i)Find the coordinates of A and B. ˉ(ii)Find the distance between A and B. ˉ(iii)Find the sum of the length of AP and BP.

更新2:

(e) Express the included angle θ of AP and BP in term of a, b and t. (f) Find the rate of change of θ when P reaches the original position.

更新3:

更正，是： x = a cos t y = b sin t

回答 (1)

The Little Prince of Mathematics

2007-05-05 6:54 pm

✔ 最佳答案

a)
Without Loss of Generality, Assume the P is at (a,0),
a = a cost and 0 = b sint
cost = 1 and sint = 0
t = 0, 2pi, 4pi, ... and t = 0, pi, 2pi, ...
Therefore combining the result, the time taken of the first time that P reaches the original position is 2 pi, where pi = 3.14159...

b)
dx/dt = -asint, dy/dt = bcost.
speed = [sqrt (dx/dt)^2+(dy/dt)^2] = b

參考: me

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