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Find the corresponding rectangular equation for the curve represented by the parametric equations given by x = 1 + 2 sin t and y = 3 cos t - 1 by eliminating the parameter.




a. 4x^2 + 9y^2 - 8x + 18y - 23 = 0

b. 9x^2 + 4y^2 - 18x + 8y - 23 = 0

c. 4x^2 + 9y^2 + 8x - 18y - 23 = 0

d. 9x^2 + 4y^2 + 18x - 8y - 23 = 0

e. 4x^2 + 9y^2 - 8x - 18y - 23 = 0

f. 9x^2 + 4y^2 - 18x - 8y - 23 = 0

or none of these

2007-06-11 07:18:26 · 2 answers · asked by Anonymous in Science & Mathematics Mathematics

2 answers

Solve the first equation for t

t = invsin((x - 1)/2)

imagine a triangle whose hypotenuse is 2 and opposite leg is (x - 1). This will give an adjacent side of sqrt(4 - (x-1)^2).

So, cos(t) = sqrt(4 - (x-1)^2) /2

y = 3 cos(t) - 1 = 3 sqrt(4 - (x-1)^2) /2 -1

y + 1 = 3/2 sqrt(4 -x^2 +2x -1)

2y + 2 = 3 sqrt(3 - x^2 + 2x)

4y^2 + 8y + 4 = 9(3 - x^2 + 2x)

9x^2 + 4y^2 -18x + 8y -23 = 0

I get b)

2007-06-11 07:32:35 · answer #1 · answered by tbolling2 4 · 1 0

x = 1 + 2 sin t and y = 3 cos t - 1
(x-1)/2 = sint
(y+1)/3 = cost
(x-1)^2/4 + (y+1)^2/9 = sin^2t +cos^2t = 1
This is the equation of an ellipse.
If you expand it you get:
(x^2-2x+1)/4 + (y^2+2y+1)/9 = 1
9x^2 -18x +9 +4y^2 +8y + 4= 36
9x^2 +4y^2 -18x +8y -23 =0
So the anwer is b.

2007-06-11 14:39:35 · answer #2 · answered by ironduke8159 7 · 1 0

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