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Question
What conic section is represented by the equation 3 x squared plus 2 y squared minus 24 x plus 30 equals 0? Explain.
The equation given, 3x^2 + 2y^2 - 24x + 30 = 0, represents an ellipse.
To determine the conic section, we can analyze the quadratic terms in the equation.
The coefficient of x^2 (3) and the coefficient of y^2 (2) have the same sign, which indicates that the ellipse is elongated along both the x-axis and y-axis simultaneously.
Also, since the coefficients of x^2 and y^2 are different, the ellipse is non-circular.
The equation of an ellipse can be written in the general form:
Ax^2 + By^2 + Cx + Dy + E = 0
Comparing this with the given equation, we find:
A = 3, B = 2
Since both A and B are positive, this suggests that the ellipse is oriented in the positive x and y directions.
Based on these observations, we can conclude that the equation 3x^2 + 2y^2 - 24x + 30 = 0 represents an ellipse.