Problem 31 - Algebra and Trigonometry

Question

Identify each equation without applying a rotation of axes. $$5 x^{2}-2 x y+5 y^{2}-12=0$$

Step-by-Step Solution

Answer

The given equation represents an ellipse according to the discriminant analysis.

Step 1: Identify the Form of the Equation

An equation in the form of \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) represents a conic section if \(B^2 - 4AC < 0\) implies an ellipse, \(B^2 - 4AC > 0\) a hyperbola, and \(B^2 - 4AC = 0\) a parabola. The given equation is \(5x^2 - 2xy + 5y^2 - 12 = 0\). Here, \(A = 5\), \(B = -2\), and \(C = 5\).

Step 2: Calculate Discriminant

The discriminant is calculated as \(B^2 - 4AC = (-2)^2 - 4*5*5 = 4 - 100 = -96\). Because the discriminant is less than zero, the equation represents an ellipse.

Step 3: Express Equation in Standard Form

We don't need to express the equation in the standard form of an ellipse because the exercise specifically asks not to apply a rotation of axes. Hence, the given equation \(5x^2 - 2xy + 5y^2 - 12 = 0\) is identified as an ellipse without applying the rotation of axes.