Recall that the methods discussed in this lesson to identify conic sections are based on using the general form equation, Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Therefore, be sure to rewrite the given equation as the general form if necessary. Make sure you are using the appropriate method (B = 0 or B ≠ 0). After you identify the conic section, click Check Your Answer to see a detailed solution to the problem.

Identify the conic section represented by each equation.

1. 8x² + 12x + 8xy = -2y² - 13y + 20
   
   
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   Check Your Answer

   Step 1: Write the equation in general form.

   8x² + 8xy + 2y² + 12x + 13y - 20 = 0

   Step 2: Identify the values of A, B and C.

   A = 8, B = 8, C = 2

   Step 3: Since B ≠ 0, use the Discriminant Formula.

   B² - 4AC
   8² - 4(8)(2) = 0

   Since B² - 4AC = 0 then the conic section is a parabola.

   
2. 3x² + 5xy + 3y² + 14x - 9y - 24 = 0
   
   
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   Step 1: Write the equation in general form.

   The equation is already in general form so it does not change.

   3x² + 5xy + 3y² + 14x - 9y - 24 = 0

   Step 2: Identify the values of A, B and C.

   A = 3, B = 5, C = 3

   Step 3: Since B ≠ 0, use the Discriminant Formula.
   
   B² - 4AC.
   3² - 4(3)(5) < 0

   Since B² - 4AC < 0 then the conic section is an ellipse.

   
3. 4y² - 8y + 3 = 9x²
   
   
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   Check Your Answer

   Step 1: Write the equation in general form.

   The equation is already in general form so it does not change.

   -9x² + 4y² - 8y + 3 = 0

   Step 2: Identify the values of A, B and C.

   A = -9, B = 0, C = 4

   Step 3: Since B = 0, use A and C to determine the conic section.

   AC = -36
   AC < 0

   Since AC < 0 then the conic section is a hyperbola.