A curve is described by the equations {x = 2 + 2 cos t and y = 1 – 3 sin^2 t} with 0 ≤ t < 2π}.?
Find an equation in x and y for this curve.
The solution that the book gives is y = 3/4x^2 - 3x + 1.
I know that the graph gives a section of a parabola, but I would like to know how they got that answer and the steps that it took to answer it. A thorough explanation is greatly appreciated.
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x = 2 + 2cost
x - 2 = 2cost ................ [1]
y = 1 - 3sin²t
y = 1 - 3(1 - cos²t)
y = 1 - 3 + 3cos²t
y = -2 + 3cos²t
y + 2 = 3cos²t ................... [2]
[1]²:
(x - 2)² = 4cos²t .................. [3]
[3] / [2]:
(x - 2)²/(y + 2) = 4/3
3(x - 2)² = 4(y + 2)
3x² - 12x + 12 = 4y + 8
4y = 3x² - 12x + 4
y = (3/4)x² - 3x + 1