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The Circle

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The Parabola

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General Form

Intersections of Lines & Conics

Summary&Test

 

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 UNIT 10  : THE CONICS

 LESSON 6: GENERAL FORM OF THE EQUATION OF A CONIC: ax2 + by2 +2gx + 2fy + c = 0

                                                                            

The type of conic can be determined by the values of the coefficients “a and b” in the equation according to the following chart.

 

                                               

Type of Conic

            Condition on a, b

Circle

a = b

Parabola

ab = 0

Ellipse

ab > 0

Hyperbola

ab < 0

 

Example 1:

Determine the type of conic in each case by completing the chart.

 

                       

Conic in general form

Value of “a”

Value of “b”

Condition on a, b

Type of Conic

2x2 + y2+ 4x + 8y – 10 = 0

 

 

 

 

x2 + y2 – 4x –16 = 0

 

 

 

 

y2 – 2x – 6y – 4 = 0

 

 

 

 

4x2 - 9y2 – 10y –14 = 0

 

 

 

 

2x2 –6x – 5y –7 = 0

 

 

 

 

 

Solution:

 

           

Conic in general form

Value of “a”

Value of “b”

Condition on a, b

Type of Conic

2x2 + y2+ 4x + 8y – 10 = 0

2

1

ab > 0

Ellipse

x2 + y2 – 4x –16 = 0

1

1

a = b

Circle

y2 – 2x – 6y – 4 = 0

0

1

ab = 0

Parabola

4x2 - 9y2 – 10y –14 = 0

4

-1

ab < 0

Hyperbola

2x2 –6x – 5y –7 = 0

2

0

ab = 0

Parabola

 

 

 

Example 2:

Given the conic with equation in general form  2x2 + 4x – 2y = 0.

      a)   Identify the conic using the tables above.

b)      Write the equation in standard form.

c)      Determine the center or vertex.

d)      Draw the graph.

       

 

 

 

 

 

 

 

      

 

 

 

 

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