Question

What is the rectangular equivalence to the parametric equations?

x(θ)=4cosθ+2,y(θ)=2sinθ−5 , where 0≤θ<2π .

Answer 1

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.

What is a parametric equation?

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.

What is the calculation for the above?

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

$\left( \frac{x-2}{n}\right)^{2}$ = cos (θ)²;        $\left( \frac{y+5}{n}\right)^{2}$ = sin(θ)²

Recall that

cos (θ)² + sin(θ)² = 1

Therefore,

$\left( \frac{x-2}{n}\right)^{2}$  + $\left( \frac{y+5}{n}\right)^{2}$ = 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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