What is the rectangular equivalence to the parametric equations?
x(θ)=4cosθ+2,y(θ)=2sinθ−5 , where 0≤θ<2π .
___+___ = 1, where x is on the interval ___
Options: (x-2)^2 /16, (x-2)^2 /16, (x-5)^2 / 4, (x+5)^2 /4, (y-2)^5/16,
(y+2)^5 /16, (y-5)^2 /4, (y+5)^2 /4, [-7,-3],[-4,4]. [-2,6], [-2,2]
(x-2)/4 = cosθ
(y+5)/2 = sinθ
now, since cos^2θ + sin^2θ = 1, ...
x = 4cosθ + 2 ---> cosθ = (x-2)/4
y = 2sinθ - 5 -----> sinθ = (y+5)/2
we know sin^2 θ + cos^2 θ = 1
(y+5)^2 /4 + (x-2)^2/16 = 1
(x-2)^2/16 + (y+5)^2 /4 = 1
an ellipse with centre at (2, -5) , |a| = 4, |b| = 2
check:
https://www.wolframalpha.com/input/?i=plot+x%3D4cos%CE%B8%2B2%2C+y%3D2sin%CE%B8%E2%88%925
To find the rectangular equivalence to the given parametric equations, we need to eliminate the parameter θ and express x and y solely in terms of each other.
From the given parametric equations:
x(θ) = 4cosθ + 2
y(θ) = 2sinθ - 5
We can solve for cosθ and sinθ in terms of x and y as follows:
cosθ = (x - 2) / 4
sinθ = (y + 5) / 2
Now we can square both sides of the equations for cosθ and sinθ:
cos^2θ = (x - 2)^2 / 16
sin^2θ = (y + 5)^2 / 4
Adding these two equations together, we get:
cos^2θ + sin^2θ = (x - 2)^2 / 16 + (y + 5)^2 / 4
Since cos^2θ + sin^2θ = 1, we have:
1 = (x - 2)^2 / 16 + (y + 5)^2 / 4
This is the rectangular equivalence to the given parametric equations.
So, the correct option is:
(x - 2)^2 / 16 + (y + 5)^2 / 4 = 1,
where x is on the interval [-4, 6].
To find the rectangular equivalence to the given parametric equations, we can substitute the values of x(θ) and y(θ) into the equation ___ + ___ = 1 and solve for x.
From the given equations, we have:
x(θ) = 4cosθ + 2
y(θ) = 2sinθ - 5
Substituting these values in the equation ___ + ___ = 1, we get:
(4cosθ + 2) + (2sinθ - 5) = 1
Now, let's simplify this equation:
4cosθ + 2 + 2sinθ - 5 = 1
4cosθ + 2sinθ - 3 = 1
To isolate the x-term, we need to express cosθ in terms of x and sinθ in terms of y. We can use the trigonometric identity:
cos²θ + sin²θ = 1
Dividing the above identity by cos²θ, we get:
1 + (sinθ / cosθ)² = (1 / cos²θ)
Rearranging the terms, we have:
(1 / cos²θ) = 1 / [1 + (sinθ / cosθ)²]
Now, let's substitute for cosθ and sinθ using the given equations:
cosθ = (x - 2) / 4
sinθ = (y + 5) / 2
Substituting these values into (1 / cos²θ), we get:
1 / cos²θ = 1 / [1 + ((y + 5) / 2)²]
Now, let's substitute this expression back into the equation 4cosθ + 2sinθ - 3 = 1:
4[(x - 2) / 4] + 2[(y + 5) / 2] - 3 = 1
Simplifying the above equation:
x - 2 + y + 5 - 3 = 1
x + y = 3
So, the rectangular equivalence to the given parametric equations is x + y = 3.
Therefore, the correct option from the given choices is (x + y = 3).
Note: The options you provided are not correct, as there is no matching option.