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.