What conic section is represented by the equation \( 3 x^{2}+2 y^{2}-24 x+30=0 \) ? Explain. (4 points) B \( I \) U

To determine the conic section represented by the equation \( 3x^{2} + 2y^{2} - 24x + 30 = 0 \), we first rearrange the equation and complete the square. The quadratic terms can be grouped as \( 3(x^{2} - 8x) + 2y^{2} + 30 = 0 \). Completing the square for \( x^{2} - 8x \) gives us \( (x - 4)^{2} - 16 \). Thus, the equation simplifies to: \[ 3((x - 4)^{2} - 16) + 2y^{2} + 30 = 0 \] This leads to: \[ 3(x - 4)^{2} + 2y^{2} - 18 = 0 \] Rearranging gives: \[ 3(x - 4)^{2} + 2y^{2} = 18 \] Dividing through by 18 yields the standard form: \[ \frac{(x - 4)^{2}}{6} + \frac{y^{2}}{9} = 1 \] This is the equation of an ellipse. The focal point of an ellipse can make for a delightful plot twist, much like a good mystery novel! Imagine the ellipse as a stretched-out circle; its center is the focal heart of the shape, providing a perfect backdrop for our drama. You can find the foci by using the distance \( c = \sqrt{a^2 - b^2} \) where \( a^2 = 9 \) and \( b^2 = 6 \). This means you can chase down those foci at \( (4, 3) \) and \( (4, -3) \) on the graph. Grab a pencil and have fun plotting—it’s all about the journey through curves!