Is it true that the equations r=8, x2+y2=64, and x=8sin⁡(3t),y=8cos⁡(3t)(0≤t≤2π) all have the same graph.

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Aubree Mcintyre · Accepted Answer

2.Now, we will check that what type of equation is the r=8 This equation is the polar form equation in which we consider r=x2+y2 So, r=x2+y2=8 ⇒=x2+y2=82=64 Thus, we can say that after changing polar co-ordinates to Cartesian co-ordinates, equation  r=8  is also the equation of the circle with radius a=8 and center at (0,0). So, the graphs of the equation  r=8 and x2+y2=64 are same which is given above. Now, we have the parametric form equations given as: x=8sin⁡(3t),y=8cos⁡(3t),0≤t≤2π x2=64sin2(3t) and y2=64cos2(3t) ⇒x2+y2=64sin2(3t)+64cos2(3t) =64(sin2(3t)+cos2(3t))=64 as by trigonometry identity sin2(3t0+cos2(3t)=1 thus, we have shown that these parametric equations from circle with radius 8 and center at (0,0). So, these parametric equations x=8sin⁡(3t),y=8cos⁡(t),0≤t≤2π also have the same graph given above. 4. So, the given statement is true as the equation  r=8 , x2+y2=64 and x=8sin⁡(3t),y=8cos⁡(t),0≤t≤2π denote same equation of circle with radius 8 and center at origin.Thus, all these equations have the same graph given above. Answer: it`s TRUE

Is it true that the equations r=8, x^2+y^2=64, and x=8sin(3t), y=8cos(3t) (0<=t<=2pi) all have the same graph.

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