Find the rectangular-coordinate equation for the parametric equations given. $$x = 2 \cos ^ { 2 } t , y = 2 \sin ^ { 2 } t$$
A) $x + y = 2,0 \leq x \leq 2$
B) $x ^ { 2 } + y ^ { 2 } = 4 , \quad 0 \leq x \leq 4$
C) $2 x ^ { 2 } + 2 y ^ { 2 } = 1 , \quad 0 \leq x \leq 2$
D) $x ^ { 2 } + y ^ { 2 } = 2 , \quad 0 \leq x \leq 2$
E) $x ^ { 2 } - y ^ { 2 } = 4$

Question

Quizplus · Accepted Answer

Using the identity $\sin^2 t + \cos^2 t = 1$, we can add the given equations: $x = 2\cos^2 t$ and $y = 2\sin^2 t$ to get $x + y = 2\cos^2 t + 2\sin^2 t = 2(\cos^2 t + \sin^2 t) = 2$. The range for $x$ and $y$ is derived from the maximum and minimum values of $\sin^2 t$ and $\cos^2 t$, which are 0 and 1, respectively, thus $0 \leq x, y \leq 2$.

Find the Rectangular-Coordinate Equation for the Parametric Equations Given x=2cos⁡2t,y=2sin⁡2tx = 2 \cos ^ { 2 } t , y = 2 \sin ^ { 2 } tx=2cos2t,y=2sin2t

Find the Rectangular-Coordinate Equation for the Parametric Equations Given $x = 2 \cos ^ { 2 } t , y = 2 \sin ^ { 2 } t$