Understanding Conic Section Identification

To identify the type of conic section represented by a general second degree equation, we use the coefficients of the quadratic terms.

The general second degree equation of a conic section is given by:

\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)

The given equation is:

\(x^2 + xy + 2y^2 + x + y = 1\)

First, we rewrite the given equation in the standard form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) by moving all terms to one side:

\(x^2 + xy + 2y^2 + x + y - 1 = 0\)

Comparing this with the general second degree equation, we can identify the coefficients:

• \(A = 1\) (coefficient of \(x^2\))
• \(B = 1\) (coefficient of \(xy\))
• \(C = 2\) (coefficient of \(y^2\))
• \(D = 1\) (coefficient of \(x\))
• \(E = 1\) (coefficient of \(y\))
• \(F = -1\) (constant term)

Using the Discriminant for Conic Section Identification

The type of conic section (non-degenerate) is determined by the value of the discriminant, which is calculated as \(B^2 - 4AC\).

• If \(B^2 - 4AC > 0\), the conic is a hyperbola.
• If \(B^2 - 4AC = 0\), the conic is a parabola.
• If \(B^2 - 4AC < 0\), the conic is an ellipse.

Let's calculate the discriminant for the given equation using the coefficients \(A=1\), \(B=1\), and \(C=2\).

Discriminant \( = B^2 - 4AC \)

\( = (1)^2 - 4(1)(2) \)

\( = 1 - 8 \)

\( = -7 \)

Classifying the Type of Conic

The calculated discriminant is \(-7\). Since \(B^2 - 4AC = -7\), which is less than 0, the equation represents an ellipse (assuming it is not a degenerate case like a point or imaginary ellipse, which is not among the options).

This method provides a straightforward way for conic section identification based purely on the coefficients of the second degree terms in the general second degree equation.

Thus, the type of conic represented by \(x^2 + xy + 2y^2 + x + y = 1\) is an ellipse, according to the rules of analytic geometry for classifying conic sections.

Therefore, the correct classification based on conic section identification using the discriminant is an ellipse.