SOLUTION: For which values of a does the conic 2x^2 + 5y^2 − 30y + 8x = a have at least one point? (Enter your answer using interval notation.)

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: For which values of a does the conic 2x^2 + 5y^2 − 30y + 8x = a have at least one point? (Enter your answer using interval notation.)      Log On


   

Question 1092186: For which values of a does the conic
2x^2 + 5y^2 − 30y + 8x = a
have at least one point? (Enter your answer using interval notation.)
Found 2 solutions by Alan3354, greenestamps:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
1 point?
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All conics have an infinite # of points.

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!

The conic is an ellipse, because it has x squared and y squared terms that have the same sign and different coefficients.

Complete the square in both variables....

2x%5E2+%2B+5y%5E2+-+30y+%2B+8x+=+a
2%28x%5E2%2B4x%29%2B5%28y%5E2-6y%29+=+a
2%28x%5E2%2B4x%2B4%29%2B5%28y%5E2-6y%2B9%29+=+a%2B2%284%29%2B5%289%29
2%28x%2B2%29%5E2%2B5%28y-3%29%5E2+=+a%2B53

The graph will be an ellipse if a+53 is positive.
The graph will be a single point if a+53 is equal to 0.
The graph will have no points if a+53 is negative.

So the graph will have at least one point if a+53 is greater than or equal to 0; i.e., if a is greater than or equal to -53.

Answer in interval notation: [-53, infinity)