What is the rectangular equivalence to the parametric equations?
x(θ)=4cosθ+2,y(θ)=2sinθ−5 , where 0≤θ<2π .

The rectangular equivalence of the parametric equations is given as:
an ellipse that is focalized on (2, -5) with semi-major axis of 4 and semi-minor axis of 2.
A parametric equation is the type which has dependent variables defined as a continuous function of the independent factor - t, and the dependent variables are mutually exclusive of other existing variables.
Step 1 - Note that we have the following terms:
x(θ)=4cosθ+2,
y(θ)=2sinθ−5
Step 2 - Rewrite the above equations
x = 4 -cos(θ) +2;
y = 2 -Sin (θ) - 5
We rewrite again as
[tex]\left( \frac{x-2}{n}\right)^{2}[/tex] = cos (θ)²; [tex]\left( \frac{y+5}{n}\right)^{2}[/tex] = sin(θ)²
Recall that
cos (θ)² + sin(θ)² = 1
Therefore,
[tex]\left( \frac{x-2}{n}\right)^{2}[/tex] + [tex]\left( \frac{y+5}{n}\right)^{2}[/tex] = 1
The result is an ellipse with the center (2, -5) with a semi-major axis of 4 and a semi-minor axis of 2. See the result of the Parametric Graph Plot attached.
Learn more about parametric equations at:
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