Identify the conic: 16x² + 9y² – 36y – 108 = 0

A. Ellipse
Correct! Both squared variables have different positive coefficients. That fact means that this equation represents an ellipse.

B. Circle
Incorrect. The coefficients for the x squared term and the y squared term must be the same to have a circle. These coefficients are different.

C. Hyperbola
Incorrect. One of the coefficients for the x squared term and the y squared term must be negative to have a hyperbola. These coefficients are both positive.

D. Parabola
Incorrect. There can only be one squared variable to have a parabola. This equation contains both an x squared term and a y squared term.

Identify the conic: x² + 4x + y² – 2y + 5 = 0

A. Ellipse
Incorrect. In an ellipse, both squared variables have different positive coefficients. In this equation, both squared variables have the same coefficient.

B. Circle
Correct! The coefficients for the x squared term and the y squared term are the same so this equation is a circle.

C. Hyperbola
Incorrect. One of the coefficients for the x squared term and the y squared term must be negative to have a hyperbola. These coefficients are both positive.

D. Parabola
Incorrect. There can only be one squared variable to have a parabola. This equation contains both an x squared term and a y squared term.

Identify the conic: 9x² - 9y² + 36x – 45 = 0

A. Ellipse
Incorrect. For an ellipse, both squared variables have positive coefficients. In this equation, the squared variables have opposite signs.

B. Circle
Incorrect. For a circle, both squared variables have positive coefficients. In this equation, the squared variables have opposite signs.

C. Hyperbola
Correct! One of the coefficients for the x² term and the y²term must be negative to have a hyperbola. This event happens here because y² has a negative coefficient.

D. Parabola
Incorrect. There can only be one squared variable to have a parabola. This equation contains both an x squared term and a y squared term.

Identify the conic 4y² – 8y + 3x – 16 = 0

A. Ellipse
Incorrect. For an ellipse, both variables must have a squared term. In this equation, there is no x² term.

B. Circle
Incorrect. For a circle, both variables must have a squared term. In this equation, there is no x² term.

C. Hyperbola
Incorrect. For a hyperbola, both variables must have a squared term. In this equation, there is no x² term.

D. Parabola
Correct! There can only be one squared variable to have a parabola and this equation contains only a y² term.