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The conics menu is very useful for dealing with conics - a special family of curves which includes parabolas, circles, ellipses and hyperbolas. There is a separate conics menu; however, graphing and analysing conics can be done through the Main program by selecting the conics icon under the graphing drop-down list:
Figure 1 - The location of the conics program in the Main Menu.
Introduction to conics
Conics are cross-section of a double-napped cone (two cones end to end):
Figure 2 - A diagram of a double-napped cone and cross sections defining conics.
As such, only certain curves and lines are conics (shown above). Others, for example straight lines and truncus, are not conic and will not work properly in the conics program on CAS.
The Conics Menu:
Commands:
When graphing a conic it will be able to show the asymptotes, something that is not possible in the regular graphing screen.
For a conic to be graphed it must be in the form y = f(x), not just f(x) which was sufficient in the past.
Under the conics analysis → G-Solve there are more option specific to conics, including (but not limited to):
- Vertex
- Centre
- Radius
- Asymptotes
These commands are demonstrated in the worked examples below:
Figure 3 - The analysis options in the Conics menu.
Worked Example 1: Parabola
A parabola [insert link] is a conic that contains a vertex. Consider the parabola y = (x - 4)2 + 3:
Figure 4 - The CAS input for the parabola y = (x - 4)2 + 3.
Vertex
To find the conics vertex use "Vertex" from the G-Solve list:
Figure 5 - The vertex command in action.
Therefore, the coordinate of the vertex is (4,3).
This parabola also has a y-intercept which is demonstrated in Worked Example 3 (below).
Worked Example 2: Hyperbola
A hyperbola [insert link] is a conic that contains asymptoes. Consider the parabola:
Figure 6 - The CAS input for the hyperbola.
Asymptotes
To find the conics asymptotes use "Asymptotes" from the G-Solve list:
Figure 7 - The asymptotes command in action.
Therefore, the equations of the asymptotes are y = -1 and x = 2.
This hyperbola also has x- and y-intercepts which are demonstrated in Worked Example 3 (below).
Worked Example 3: Circles
A circle [insert link] is a conic that contains a centre and radius, and depending on the equation x- and y-intercepts. Consider the circle:
Figure 8 - The CAS input for the circle.
Centre
To find the conics centre use "Centre" from the G-Solve list:
Figure 9 - The centre command in action.
Therefore, the coordinates of the centre of the circle are (-1,2).
Radius
To find the conics radius use "Radius" from the G-Solve list:
Figure 10 - The radius command in action.
Therefore, the radius of the circle is 4 units.
y-intercepts
If you attempted to find the y-intercepts of a circle in the normal graphing screen it would not be able to as a circle is not a function [insert link] (fails the vertical line test [insert link]). However, in conics it is easily able to:
To find the conics y-intercepts use "y-intercept" from the G-Solve list:
Figure 11 - The y-intercept command in action.
Using the "left" and "right" arrows on the hard keyboard allows you to find each y-intercept. Therefore, the numerical coordinates are (0,-1.873) and (0,5.873). Remember, exact answers are generally required in VCE Mathematical Methods - to find these use the Main Menu and solve the equation for y when you let x = 0 (this gives the y-intercepts):
Figure 12 - Find the exact solutions using solve in the Main Menu.
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