Question

The equation \( 5x^2 + y^2 + y = 8 \) represents:

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

Hint:

To confirm the shape, you can complete the square to find the standard form: \( 5x^2 + (y + \frac{1}{2})^2 = 8 + \frac{1}{4} = \frac{33}{4} \). Dividing by the constant on the right gives \( \frac{x^2}{33/20} + \frac{(y + 1/2)^2}{33/4} = 1 \), which is clearly the standard equation of an ellipse.

Answer 1

The Correct Option is A

Concept: The general second-degree equation in two variables \( x \) and \( y \) is \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). The type of conic section represented depends on the relationship between the coefficients of the squared terms and the \( xy \) term. For an equation where \( B = 0 \) (no \( xy \) term), we look at coefficients \( A \) and \( C \):
• If \( A = C \), it represents a circle.
• If \( A \neq C \) but both have the same sign, it represents an ellipse.
• If \( A \) and \( C \) have opposite signs, it represents a hyperbola.
• If either \( A \) or \( C \) is zero, it represents a parabola.

Step 1: Analyze the coefficients of the given equation.
Rearrange the equation into the general form: \( 5x^2 + y^2 + y - 8 = 0 \). Comparing this to \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), we find:
• \( A = 5 \) (coefficient of \( x^2 \))
• \( B = 0 \) (no \( xy \) term)
• \( C = 1 \) (coefficient of \( y^2 \))

Step 2: Apply the classification criteria.
Since \( B = 0 \), we evaluate \( A \) and \( C \). Both coefficients are positive (same sign), which rules out a hyperbola or parabola. Furthermore, \( A = 5 \) and \( C = 1 \), so \( A \neq C \). Because the coefficients of the squared terms are different but have the same sign, the equation represents an ellipse.