Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.
x=4+2cos(theta) y=-1+2sin(theta)
cosθ = (x-4)/2
sinθ = (y+1)/2
(x-4)^2 + (y+1)^2 = 4
it's easy to see why this drops out if you rewrite the original equations as
(x-4) = 2cosθ
(y+1) = 2sinθ
So, it's the regular equation for a circle, but centered at (4,-1) instead of (0,0)
To eliminate the parameter and write the corresponding rectangular equation, we need to express x and y in terms of each other without using the parameter theta.
Given:
x = 4 + 2cos(theta)
y = -1 + 2sin(theta)
To eliminate the parameter, we can start by squaring both equations:
x^2 = (4 + 2cos(theta))^2
y^2 = (-1 + 2sin(theta))^2
Next, we can use the trigonometric identity:
cos^2(theta) + sin^2(theta) = 1
To manipulate the equations and eliminate the parameter theta, we can replace cos^2(theta) in the first equation with (1 - sin^2(theta)):
x^2 = (4 + 2(1 - sin^2(theta)))^2
Simplifying further:
x^2 = (4 + 2 - 2sin^2(theta))^2
x^2 = (6 - 2sin^2(theta))^2
x^2 = 36 - 24sin^2(theta) + 4sin^4(theta) ----(Equation 1)
Next, we can replace sin^2(theta) in the second equation with (1 - cos^2(theta)):
y^2 = (-1 + 2(1 - cos^2(theta)))^2
y^2 = (-1 + 2 - 2cos^2(theta))^2
y^2 = (1 - 2cos^2(theta))^2
y^2 = 1 - 4cos^2(theta) + 4cos^4(theta) ----(Equation 2)
Now, we have eliminated the parameter theta, and we can rewrite the equations in terms of x and y. By rearranging Equation 1, we get:
x^2 - 36 = -4sin^4(theta) + 24sin^2(theta)
Similarly, by rearranging Equation 2:
y^2 - 1 = 4cos^4(theta) - 4cos^2(theta)
Simplifying further, we obtain the corresponding rectangular equation:
(x^2 - 36)/24 = sin^2(theta)
(y^2 - 1)/4 = cos^2(theta)
Thus, the corresponding rectangular equation whose graph represents the curve is:
(x^2 - 36)/24 = sin^2(theta) and (y^2 - 1)/4 = cos^2(theta)
To eliminate the parameter and convert the equations in parametric form to rectangular form, you can use the following steps:
1. Start with the given parametric equations:
x = 4 + 2cos(theta)
y = -1 + 2sin(theta)
2. Square both sides of the x equation:
x^2 = (4 + 2cos(theta))^2
3. Expand the right side:
x^2 = 16 + 16cos(theta) + 4cos^2(theta)
4. Rearrange the equation to isolate cos(theta):
x^2 - 16 = 16cos(theta) + 4cos^2(theta)
5. Use the identity cos^2(theta) = 1 - sin^2(theta) to substitute for cos^2(theta):
x^2 - 16 = 16cos(theta) + 4(1 - sin^2(theta))
6. Simplify the expression:
x^2 - 16 = 16cos(theta) + 4 - 4sin^2(theta)
7. Move all terms to one side:
x^2 - 20 = 16cos(theta) - 4sin^2(theta)
8. Rewrite the y equation:
y = -1 + 2sin(theta)
9. Square both sides of the y equation:
y^2 = (-1 + 2sin(theta))^2
10. Expand the right side:
y^2 = 1 - 4sin(theta) + 4sin^2(theta)
11. Rearrange the equation to isolate sin(theta):
y^2 - 1 = -4sin(theta) + 4sin^2(theta)
12. Substitute for sin^2(theta) using the identity sin^2(theta) = 1 - cos^2(theta):
y^2 - 1 = -4sin(theta) + 4(1 - cos^2(theta))
13. Simplify the expression:
y^2 - 1 = -4sin(theta) + 4 - 4cos^2(theta)
14. Move all terms to one side:
y^2 - 5 = -4cos^2(theta) - 4sin(theta)
15. Now, combine the two equations obtained for x and y:
x^2 - 20 = 16cos(theta) - 4sin^2(theta)
y^2 - 5 = -4cos^2(theta) - 4sin(theta)
16. Rearrange the equations, combining like terms for each variable:
x^2 + 4sin^2(theta) - 16cos(theta) - 20 = 0
4cos^2(theta) + 4sin(theta) - y^2 + 5 = 0
17. The final rectangular equation eliminating the parameter for the given parametric equations is:
x^2 + 4sin^2(theta) - 16cos(theta) - 20 = 0
4cos^2(theta) + 4sin(theta) - y^2 + 5 = 0