What is the rectangular equivalence to the parametric equations?

Question

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 [  ,  ]

 

Answer choices: 

 

(x−2)² / 16  ,  (x+2)² / 16  ,  (x−5)² / 4  ,  (x+5)² / 4  ,  (y−2)⁵ / 16  ,  (y+2)² / 16  ,  (y−5)² / 4  ,  (y+5)24

 

[−7,−3]  ,  [−4,4]  ,  [−2,6]  ,  [−2,2]

 

 

 

Hello! I'm struggling with how to do this. I tried looking up multiple videos but I can't quite understand. If someone could help me, I'd appreciate that! Thanks!! :)

Answer 1

x = 4cosθ + 2 So, cosθ = (x-2) / 4

y = 2sinθ - 5 So, sinθ = (y+5) / 2

Since cos²θ + sin²θ = 1, we have (x-2)² / 16 + (y+5)² / 4 = 1

Since -1 ≤ cosθ ≤ 1, -4 ≤ 4cosθ ≤ 4. So, -2 ≤ 4cosθ + 2 ≤ 6. That is, x lies in the interval [-2,6].

Answer 2

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