## Conic Sections General Form: A x2 + B xy + C y2 + D x + E y + F = 0

• Note: When B = 0, the conic section is parallel to the x- or y-axis. (Pure Math 30 focus)
• Further Note: When B 0, the conic section is rotated relative to the x- or y-axis.

What to Learn

Help to Remember What to Learn

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) A = C 0, AC > 0 Circle gets smaller as |A| and |C| increase. 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) A or C = 0, AC = 0 Opens up/down when: C = 0 Opens left/right when: A = 0 For Pure Math 30, B = 0 = 0 hyperbola (two intersecting lines) A 0 and C 0 AC > 0 AC < 0 Opens up/down when: A < 0 Opens left/right when: C < 0 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).

A copy of the above applet is provided to study the remainder of the notes for descriptions of stretches.

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.

Parts of this work has been adapted from a Math 30 Pure learning resource originally produced and owned by Alberta Education

Comments to:  Jim Reed - Homepage