Question

The equation \( 3x^2 + 10xy + 11y^2 + 14x + 12y + 5 = 0 \) represents:

• A. a circle
• B. an ellipse
• C. a hyperbola
• D. a parabola

Hint:

The discriminant \( \Delta = B^2 - 4AC \) helps classify conic sections: - \( \Delta>0 \) indicates a hyperbola, - \( \Delta = 0 \) indicates a parabola, - \( \Delta<0 \) indicates an ellipse.

Answer 1

The Correct Option is B

The general equation of a conic is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The classification of the conic depends on the discriminant \( \Delta = B^2 - 4AC \). Here, \( A = 3 \), \( B = 10 \), and \( C = 11 \). We compute the discriminant: \[ \Delta = B^2 - 4AC = 10^2 - 4(3)(11) = 100 - 132 = -32. \] Since \( \Delta<0 \), the conic is an ellipse. Thus, the correct answer is: \[ \boxed{{an ellipse}}. \]