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1. Express the following in rectangular form.

x = 3 sin t , y = 2 cos t

2. Express the following in rectangular form:

x = 3t + 6 , y = -2t + 4

Respuesta :

sqdancefan

Answer:

  1. (x/3)² +(y/2)² = 1
  2. 2x +3y -24 = 0

Step-by-step explanation:

You want the given parametric equations expressed in rectangular form.

  1. x = 3sin(t); y = 2cos(t)
  2. x = 3t +6; y = -2t +4

Parametric equations

In general, you want to find the parameter value in terms of x and y, or make use of a relation between x and y that eliminates the parameter.

1. Ellipse

We can eliminate the parameter by making use of a trig relationship between sine and cosine.

  x/3 = sin(t)
  y/2 = cos(t)

The trig relation is ...

  sin(t)² +cos(t)² = 1

Using the above expressions for sin(t) and cos(t), we have ...

  (x/3)² +(y/2)² = 1 . . . . . . . the rectangular equation for the ellipse

2. Line

Solving each equation for t gives ...

  t = (x -6)/3
  t = (y -4)/(-2)

Setting these equal gives the equation for a line:

  (x -6)/3 = (y -4)/(-2) . . . . . . . equate expressions for t

  2(x -6) = -3(y -4) . . . . . . . . . multiply by 6

  2(x -6) +3(y -4) = 0 . . . . . . . add 3(y -4)

  2x +3y -24 = 0 . . . . . general form equation for the line

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