SOLUTION: #9 Identify x = 4sint y = 5cost a) circle b) parabola c) ellipse d) hyperbola e) none of these

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Question 122072: #9
Identify x = 4sint
y = 5cost

a) circle
b) parabola
c) ellipse
d) hyperbola
e) none of these

Answer by jim_thompson5910(28504) About Me  (Show Source):
You can put this solution on YOUR website!
x=4%2Asin%28t%29 Start with the first parametric equation

x%2F4=sin%28t%29 Divide both sides by 4 to isolate sin%28t%29.


y=5%2Acos%28t%29 Start with the second parametric equation

x%2F5=sin%28t%29 Divide both sides by 5 to isolate cos%28t%29.
Now we're going to use the trig identity:

%28sin%28t%29%29%5E2%2B%28cos%28t%29%29%5E2=1

%28x%2F4%29%5E2%2B%28y%2F5%29%5E2=1 Replace sin%28t%29 with x%2F4. Replace cos%28t%29 with y%2F5. This is why we isolated sine and cosine.

x%5E2%2F16%2By%5E2%2F25=1 Square x%2F4 to get x%5E2%2F16. Square y%2F5 to get y%5E2%2F25.



Notice how the equation is now in the form %28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1 (note: h and k are equal to zero in this case)


So this shows us that x%5E2%2F16%2By%5E2%2F25=1 graphs an ellipse (since the above equation is the general equation of an ellipse).

So this means that the two parametric equations also graph an ellipse. So the answer is C)