To identify the conic section represented by the equation \( 4y^2 - 9x^2 - 48y - 18x + 99 = 0 \), we can rearrange the equation and look for specific characteristics of the conic.
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Rearranging the equation: \[ 4y^2 - 48y - 9x^2 - 18x + 99 = 0 \]
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Group the \(y\) terms and \(x\) terms: \[ 4(y^2 - 12y) - 9(x^2 + 2x) + 99 = 0 \]
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Complete the square for the \(y\) terms: \[ y^2 - 12y = (y - 6)^2 - 36 \implies 4((y - 6)^2 - 36) = 4(y - 6)^2 - 144 \]
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Complete the square for the \(x\) terms: \[ x^2 + 2x = (x + 1)^2 - 1 \implies -9((x + 1)^2 - 1) = -9(x + 1)^2 + 9 \]
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Substituting back into the equation: \[ 4(y - 6)^2 - 144 - 9(x + 1)^2 + 9 + 99 = 0 \] Simplifying gives: \[ 4(y - 6)^2 - 9(x + 1)^2 - 36 = 0 \] \[ 4(y - 6)^2 - 9(x + 1)^2 = 36 \]
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Dividing by 36: \[ \frac{4(y - 6)^2}{36} - \frac{9(x + 1)^2}{36} = 1 \] \[ \frac{(y - 6)^2}{9} - \frac{(x + 1)^2}{4} = 1 \]
This is the standard form of a hyperbola, which is given by the equation: \[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \]
where \(a^2 = 9\) and \(b^2 = 4\).
Thus, the correct response is:
hyperbola.