We have to mention that the given equations are called
parametric equations:
x = f(t) and y = g(t), where t is the
parameter.
x = 2 + 2 cos t
(1)
y = 1 – 3 sin^2 t (2)
To
determine the equation of the line, we'll have to eliminate the parameter
t.
Since the equation for y contains the term 3 sin^2 t,
we'll try to obtain the same term, but with opposite sign, in the equation for
x.
The first step will be to isolate the term in t to the
right side and to square raise, both sides, the equation
(1):
(x-2)^2 = (2 cos
t)^2
We'll expand the square from the right
side:
x^2 - 4x + 4 = 4 (cos t)^2
(3)
We'll isolate 3(sin t)^2 to the right side, in
equation (2):
1 - y = 3(sin t)^2
(4)
We'll multiply (3) by 3 and (4) by
4:
3x^2 - 12x + 12 = 12 (cos t)^2
(5)
4 - 4y = 12(sin t)^2
(6)
We'll add (5)+(6):
3x^2 -
12x + 12+4 - 4y=12 (cos t)^2+12(sin t)^2
We'll factorize by
12 to the right side:
3x^2 - 12x + 12+4 - 4y=12[(sin t)^2 +
(cos t)^2]
From the fundamental formula of trigonometry,
we'll have:
(sin t)^2 + (cos t)^2 =
1
3x^2 - 12x + 12+4 -
4y=12
We'll subtract 12 both
sides:
3x^2 - 12x + 12 + 4 - 4y - 12 =
0
We'll combine and eliminate like
terms:
3x^2 - 12x - 4y + 4 =
0
We'll add 4y both sides and we'll use symmetric
property:
4y = 3x^2 - 12x +
4
We'll divide by 4:
y
= 3x^2/4 - 3x + 1
The equation of the line
described by the parametric equations, x = f(t) and y = g(t),
is:
y = 3x^2/4 - 3x +
1
No comments:
Post a Comment