What should be the value of k for the equation 2x^2 + 3x + 2y^2 - 5y + k = 0 to represent a circle.
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The general form of the equation of a circle with center (a, b) and radius r is (x - a)^2 + (y - b)^2 = r^2.
2x^2 + 3x + 2y^2 - 5y + k = 0
=> x^2 + 3x/2 + y^2 - 5y/2 + k/2 = 0
=> x^2 + 3x/2 + 9/16 + y^2 - 5y/2 + 25/16 + k/2 = 9/16 + 25/16
=> x^2 + 3x/2 + 9/16 + y^2 - 5y/2 + 25/16 = 34/16 - k/2
=> (x + 3/4)^2 + (y - 5/4)^2 = 34/16 - k/2
For the equation to represent a circle 34/16 - k/2 > 0
=> 34/16 > k/2
=> 34/8 > k
The value of k should be less than 17/4.
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