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Chapter 10: Problem 31

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

Short Answer

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

Step by step solution

01

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 \(C = 5\).
02

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.
03

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.

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