(a) Modifying the parametric equations of a unit circle, �find parametric equations for the ellipse:
x^2/a^2 + y^2/b^2 = 1
(b) Eliminate the parameter to �find a Cartesian equation of the curve
x=2sint; y=4+cost; t >(or equal to) 0
(a) x = a sint y = b cost
(b) (x)^2/4 + (y-4)^2 = 1
can you explain how you got both answers?
(a) To find the parametric equations for the ellipse x^2/a^2 + y^2/b^2 = 1, we can use the identity cos^2(t) + sin^2(t) = 1.
Let's choose a parameter t for the unit circle equations x = cos(t) and y = sin(t). We need to scale these equations to match the ellipse.
For x, we have x = a * cos(t) since the unit circle's radius is 1 and the ellipse's major axis is a.
For y, we have y = b * sin(t) since the unit circle's radius is 1 and the ellipse's minor axis is b.
So the parametric equations for the ellipse are:
x = a * cos(t)
y = b * sin(t)
(b) To eliminate the parameter t and find the Cartesian equation of the curve given x = 2*sin(t) and y = 4 + cos(t), we can square both equations and use the identity sin^2(t) + cos^2(t) = 1.
Multiplying the first equation x = 2*sin(t) by itself, we get x^2 = 4*sin^2(t).
Multiplying the second equation y = 4 + cos(t) by itself, we get y^2 = (4 + cos(t))^2.
Now, substitute sin^2(t) with (1 - cos^2(t)) using the identity, so x^2 = 4*(1 - cos^2(t)) = 4 - 4*cos^2(t).
Substituting this into y^2, we have y^2 = (4 + cos(t))^2 = 16 + 8*cos(t) + cos^2(t).
Now, add x^2 and y^2 together:
x^2 + y^2 = (4 - 4*cos^2(t)) + (16 + 8*cos(t) + cos^2(t)).
Simplifying this equation, we get:
x^2 + y^2 = 20 + 4*cos(t) - 3*cos^2(t).
So the Cartesian equation of the curve is:
x^2 + y^2 = 20 + 4*cos(t) - 3*cos^2(t)