Worksheet: Discriminant of a Conic Section

In this worksheet, we will practice determining the type of conic sections without rewriting the general equation in standard form, using the discriminant.

Q1:

By calculating the discriminant, identify the type of conic that is described by the equation 𝑥 + 𝑦 + 1 0 𝑥 4 𝑦 + 2 8 = 0 2 2 .

  • AA parabola
  • BA hyperbola
  • CAn ellipse
  • DA circle

Q2:

The general equation of a conic has the form 𝐴 𝑥 + 𝐵 𝑥 𝑦 + 𝐶 𝑦 + 𝐷 𝑥 + 𝐸 𝑦 + 𝐹 = 0 2 2 .

Which of the following conditions would allow us to conclude that it is an ellipse?

  • A 𝐵 4 𝐴 𝐶 > 0 2
  • B 𝐵 4 𝐴 𝐶 = 0 2
  • C 𝐵 4 𝐴 𝐶 < 0 2 , 𝐵 = 0 , and 𝐴 = 𝐶
  • D 𝐵 4 𝐴 𝐶 < 0 2 and either 𝐵 0 or 𝐴 𝐶

Q3:

The general equation of a conic has the form 𝐴 𝑥 + 𝐵 𝑥 𝑦 + 𝐶 𝑦 + 𝐷 𝑥 + 𝐸 𝑦 + 𝐹 = 0 2 2 .

Consider the equation 2 𝑥 3 𝑦 1 6 𝑥 3 0 𝑦 4 9 = 0 2 2 .

Calculate the value of the discriminant 𝐵 4 𝐴 𝐶 2 .

Hence, identify the conic described by the equation.

  • ACircle
  • BEllipse
  • CParabola
  • DHyperbola

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