|
|
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.