document.write( "Question 1092186: For which values of a does the conic
\n" ); document.write( "2x^2 + 5y^2 − 30y + 8x = a
\n" ); document.write( " have at least one point? (Enter your answer using interval notation.) \n" ); document.write( "

Algebra.Com's Answer #706757 by greenestamps(13200) $\"\"$ $\"About$

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

\n" ); document.write( "Complete the square in both variables....

\n" ); document.write( " $\"2x%5E2+%2B+5y%5E2+-+30y+%2B+8x+=+a\"$
\n" ); document.write( " $\"2%28x%5E2%2B4x%29%2B5%28y%5E2-6y%29+=+a\"$
\n" ); document.write( " $\"2%28x%5E2%2B4x%2B4%29%2B5%28y%5E2-6y%2B9%29+=+a%2B2%284%29%2B5%289%29\"$
\n" ); document.write( " $\"2%28x%2B2%29%5E2%2B5%28y-3%29%5E2+=+a%2B53\"$

\n" ); document.write( "The graph will be an ellipse if a+53 is positive.
\n" ); document.write( "The graph will be a single point if a+53 is equal to 0.
\n" ); document.write( "The graph will have no points if a+53 is negative.

\n" ); document.write( "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.

\n" ); document.write( "Answer in interval notation: [-53, infinity) \n" ); document.write( "

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