Conic
General Form (For Pure Math 30, B = 0)
Study Restricted to:
A x2 + C y2 + D x + E y + F = 0, ( B = 0)
| Conic Sections (degenerate forms) |
Parameter Restrictions
|
( ) Discriminate
B2 - 4AC
|
| circle (point or no curve) |
For Pure Math 30, B = 0 |
< 0
|
| ellipse (point or no curve) |
-
A
 C,
thus A 0
and C 0
- AC > 0
-
Vertical when:
|A| > |C|
-
Horizontal when:
|A| < |C|
For Pure Math 30, B = 0
|
< 0
|
| parabola (2 parallel lines, 1 line or no curve) |
For Pure Math 30, B = 0
|
= 0
|
| hyperbola (two intersecting lines) |
For Pure Math 30, B = 0 |
> 0
|
Note: The ellipse and hyperbola are the conic sections where A 0
and C 0.
| Algebraic Description of Stretch |
If the equation of the conic in general form is
stretched horizontally by a factor of a, substitute
x/a for
x.
If the equation of the conic in general form is stretched vertically
by a factor of b, substitute y/b for
y |
|
Interactive Activity
- Click on the applet to activate
- Set B = 0 (focus for Pure Math
30)
- Determine the parameters A, C, D, E and F (set
B = 0) needed to create circles, ellipses, parabolas and hyperbolas.
- Note: = B2 -
4AC
- Notice when B = 0, the conic is parallel
to the x- or y-axis.
- Notice when B 0,
the conic is oblique (not parallel to the x- or y-axis).
|
Convert a given
equation of a conic section from general to standard form.
Example 1
Find the domain and range of the graph of:
x2 + y2 -
4x - 6y + 13 = 0.
Solution
Because A = C 0
and B = 0, the equation represents a circle or one of its degenerate
forms.
Convert equation from general to standard form, then find the centre
and the radius of the circle. 
The centre is at (2, 3), and the radius is 0. This is an example of
a point circle.
Therefore, the domain is x = 2 and the range is y = 3.
|
Example 2
Find the domain and range of the graph of:
2x2 + 2y2 -
8x - 12y - 8 = 0.
Solution
Because A = C 0 and B = 0, the equation represents a circle or one
of its degenerate forms.
Find the centre and the radius of the circle.
Graph the circle to help determine the domain and range.
According to the graph, the domain (x-values) and range (y-values) can
be determined by adding and subtracting  to
the coordinates of the centre. Therefore, the domain is  and
the range is 
|
Example 3
Given 4x2 + 25y2 - 8x + 150y + 129 = 0, do the following.
a. Express the equation in standard form.
b. Draw the ellipse.
Solution
Note: Standard form is also called completed square form.
The centre of this ellipse is at (h, k) or (1, -3).
The major axis is parallel to the x-axis, and its endpoints are a =
5 units from the centre.
The minor axis is parallel to the y-axis, and its endpoints are b =
2 units from the centre.
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