To determine the conic section represented by the equation \( 3x^2 + 2y^2 - 24x + 30 = 0 \), we first rearrange it into a more standard form by completing the square.
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Rearranging the Equation: \[ 3x^2 - 24x + 2y^2 + 30 = 0 \] Move \( 30 \) to the right side: \[ 3x^2 - 24x + 2y^2 = -30 \]
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Completing the Square for \( x \): Factor out \( 3 \) from the \( x \) terms: \[ 3(x^2 - 8x) + 2y^2 = -30 \] Complete the square for \( x^2 - 8x \): \[ x^2 - 8x = (x-4)^2 - 16 \] Substitute back: \[ 3((x - 4)^2 - 16) + 2y^2 = -30 \] This simplifies to: \[ 3(x - 4)^2 - 48 + 2y^2 = -30 \] Combine and simplify: \[ 3(x - 4)^2 + 2y^2 = 18 \] Divide the entire equation by \( 18 \): \[ \frac{(x - 4)^2}{6} + \frac{y^2}{9} = 1 \]
This is in the standard form of an ellipse: \[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \] where \( (h, k) \) is the center and \( a^2 = 6, b^2 = 9 \). Since both \( a^2 \) and \( b^2 \) are positive and unequal, this confirms it is an ellipse.
Next, we identify the orientation based on the denominators:
- Since \( a^2 < b^2 \) (6 < 9), the major axis corresponds to the \( y \)-direction, meaning the ellipse is oriented vertically.
Conclusion: The conic section represented by the equation \( 3x^2 + 2y^2 - 24x + 30 = 0 \) is an ellipse, and it is oriented vertically in the equation \( \frac{(x-5)^2}{3} + \frac{(y+1)^2}{5} = 1 \).