jdlogo

jdlogo

jdlogo

jdlogo

jdlogo

Home

Locus

The Circle

The Ellipse

The Parabola

The Hyperbola

General Form

Intersections of Lines & Conics

Summary&Test

 

jdsmathnotes

 

 

 UNIT 9 : 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.

 

 

 

 

 

 

 

 

 

 

Return to top of page

Click here to go to homework questions