     Home Locus The Circle The Ellipse The Parabola The Hyperbola General Form Intersections of Lines & Conics Summary&Test ### 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.    