SOLUTION: Classify the conic section. (State whether it is a hyperbola, ellipse, circle or parabola). Be sure to use correct spelling or your answer will be marked wrong. 16x^2 + 9y^2 + 24x

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Classify the conic section. (State whether it is a hyperbola, ellipse, circle or parabola). Be sure to use correct spelling or your answer will be marked wrong. 16x^2 + 9y^2 + 24x       Log On


   

Question 288643: Classify the conic section. (State whether it is a hyperbola, ellipse, circle or parabola). Be sure to use correct spelling or your answer will be marked wrong. 16x^2 + 9y^2 + 24x - 36y + 23 = 0
Also my teacher said that this problem couldnt be an ellipse because I originally thought thats what this was but unfortunately its not so thats why I'm confused, please help!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
16x%5E2+%2B+9y%5E2+%2B+24x+-+36y+%2B+23+=+0+ Start with the given equation.


16x%5E2+%2B+9y%5E2+%2B+24x+-+36y++=+-23+ Subtract 23 from both sides


%2816x%5E2%2B+24x%29+%2B+%289y%5E2++-+36y%29+=+-23+ Group like terms.


16%28x%5E2%2B+%283%2F2%29x%29+%2B+9%28y%5E2++-+4y%29+=+-23+ Factor out the leading terms from each group (to make their coefficients equal to 1).


16%28x%5E2%2B+%283%2F2%29x%2B9%2F4-9%2F4%29+%2B+9%28y%5E2++-+4y%2B4-4%29+=+-23+ Complete the square for each group (let me know if you need help with that)


Group the first three terms in each group.


16%28%28x%5E2%2B3%2F4%29%5E2-9%2F16%29+%2B+9%28%28y-2%29%5E2-4%29+=+-23+ Factor each grouped trinomial.


16%28x%5E2%2B3%2F4%29%5E2-9+%2B+9%28y-2%29%5E2-36+=+-23+ Distribute.


16%28x%5E2%2B3%2F4%29%5E2+%2B+9%28y-2%29%5E2-45+=+-23+ Combine like terms.


16%28x%5E2%2B3%2F4%29%5E2+%2B+9%28y-2%29%5E2+=+-23%2B45+ Add 45 to both sides.


16%28x%5E2%2B3%2F4%29%5E2+%2B+9%28y-2%29%5E2+=+22+ Combine like terms.


%2816%28x%5E2%2B3%2F4%29%5E2%29%2F22+%2B+%289%28y-2%29%5E2%29%2F22+=+1+ Divide every term outside the parenthesis by 22 to make the right side equal to 1.


%28%28x%5E2%2B3%2F4%29%5E2%29%2F%2822%2F16%29+%2B+%28%28y-2%29%5E2%29%2F%2822%2F9%29+=+1+ Rearrange the terms.


So %28%28x%5E2%2B3%2F4%29%5E2%29%2F%2822%2F16%29+%2B+%28%28y-2%29%5E2%29%2F%2822%2F9%29+=+1+ is an equation in the form of %28%28x%5E2-h%29%5E2%29%2F%28a%5E2%29+%2B+%28%28y-k%29%5E2%29%2F%28b%5E2%29+=+1+ where h=-3%2F4, k=2, a%5E2=22%2F16 and b%5E2=22%2F9.


Recall that %28%28x%5E2-h%29%5E2%29%2F%28a%5E2%29+%2B+%28%28y-k%29%5E2%29%2F%28b%5E2%29+=+1+ is the general equation of an ellipse. So %28%28x%5E2%2B3%2F4%29%5E2%29%2F%2822%2F16%29+%2B+%28%28y-2%29%5E2%29%2F%2822%2F9%29+=+1+ is an ellipse.


Because %28%28x%5E2%2B3%2F4%29%5E2%29%2F%2822%2F16%29+%2B+%28%28y-2%29%5E2%29%2F%2822%2F9%29+=+1+ is equivalent to 16x%5E2+%2B+9y%5E2+%2B+24x+-+36y+%2B+23+=+0+, this means that 16x%5E2+%2B+9y%5E2+%2B+24x+-+36y+%2B+23+=+0+ is also an ellipse.


So 16x%5E2+%2B+9y%5E2+%2B+24x+-+36y+%2B+23+=+0+ is an ellipse.


So either there's a typo or your teacher is mistaken.