SOLUTION: 5. Identify the curve and sketch its graph. 25x^2 + 16y^2 +100x -32y – 284 = 0 (Hint: complete the square in both x and y if necessary)

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 5. Identify the curve and sketch its graph. 25x^2 + 16y^2 +100x -32y – 284 = 0 (Hint: complete the square in both x and y if necessary)       Log On


   

Question 129115: 5. Identify the curve and sketch its graph.

25x^2 + 16y^2 +100x -32y – 284 = 0
(Hint: complete the square in both x and y if necessary)
Label important features of this graph.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
25x%5E2+%2B+16y%5E2+%2B100x-32y-284+=+0 Start with the given equation


25x%5E2%2B100x%2B16y%5E2-32y-284=0 Rearrange the terms


25x%5E2%2B100x%2B16y%5E2-32y=284 Add 284 to both sides


25%28x%2B2%29%5E2-100%2B16y%5E2-32y=284 Complete the square for the x terms


25%28x%2B2%29%5E2-100%2B16%28y-1%29%5E2-16=284 Complete the square for the y terms


25%28x%2B2%29%5E2%2B16%28y-1%29%5E2-116=284 Combine like terms


25%28x%2B2%29%5E2%2B16%28y-1%29%5E2=284%2B116 Add 116 to both sides


25%28x%2B2%29%5E2%2B16%28y-1%29%5E2=400 Combine like terms



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%2825%28x%2B2%29%5E2%2B16%28y-1%29%5E2%29%2F%28400%29=%28400%29%2F%28400%29Now divide both sides by 400 to make the right side equal to 1


%2825%28x%2B2%29%5E2%29%2F%28400%29%2B%2816%28y-1%29%5E2%29%2F%28400%29=1 Simplify and break up the fraction


%28x%2B2%29%5E2%2F%2816%29%2B%28y-1%29%5E2%2F%2825%29=1 Reduce


So now the equation is in the form

%28x-h%29%5E2%2F%28a%5E2%29%2B%28y-k%29%5E2%2F%28b%5E2%29=1

which is the general equation of an ellipse. Where "a" is the horizontal diameter,"b" is the vertical diameter, and (h,k) is the center.


So in our case the horizontal diameter is a=4, the vertical diameter is b=5, and the center is (-2,1)



So if we graph %28x%2B2%29%5E2%2F%2816%29%2B%28y-1%29%5E2%2F%2825%29=1, we get