In the previous section, you investigated conic sections from the general form of the conic section equation with the restriction that B, the coefficient of the xy term, was equal to 0. When B ≠ 0, the xy term is present, and a different set of rules applies. In this section, you will use the same applet to investigate how the presence of an xy term impacts the type of conic section you will see from the general form of the equation.

Quadratic equations can be solved using the Quadratic Formula, x = negative b plus or minus the square root of b squared minus four a c all over two a -b ± √b2 − 4ac 2a when Ax2 + Bx + C = 0. Recall that the radic and, or the expression b2 - 4ac, is called the discriminant of a quadratic equation. In this section, you will learn how the discriminant of the general form of a conic section can be used to determine a conic from its equation

This activity might not be viewable on your mobile device.Interactive exercise. Assistance may be required. Below is a screen shot and link to an online applet. Click on the image to launch a new window/tab and follow the directions below. When you are done with this exercise, close the window/tab to return to this lesson.

  1. Use the sliders to set the values for A, B, and C as shown in the table below.
  2. In your notes, create and complete a table like the one shown. The first one is done for you.
  3. After you complete your table, mouse over the table above to check your answers.
  4. Answer the questions below.
Summarize Your Findings
  1. When B2 - 4AC is negative, what type of conic section results?

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    Check Your Answer

    circle or ellipse Close Pop Up
  2. When B2 - 4AC is negative, B = 0 and A = C, what type of conic section results?
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    Check Your Answer

    circle Close Pop Up
  3. When B2 - 4AC is positive, what type of conic section results?
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    Check Your Answer

    hyperbola Close Pop Up
  4. When B2 - 4AC is zero what type of conic section results?

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    Check Your Answer

    parabola Close Pop Up
Generalize the Discriminant Theorem

For the general form equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the equation generates:

  • an ellipse if
  • a circle if
  • a hyperbola if
  • a parabola if
  1. B2 − 4AC < 0 and A ≠ C
  2. B2 − 4AC < 0 and A = C and B = 0
  3. B2 − 4AC > 0
  4. B2 − 4AC = 0