Lesson 1.1. Introduction to Conic Sections
and Circles
Learning Outcomes of the Lesson
At the end of the lesson, the student is able to:
1. illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and
degenerate cases;
2. define a circle;
3. determine the standard form of the equation of a circle;
4. graph a circle in a rectangular coordinate system; and
5. solve situational problems involving conic sections (circles).
Lesson Outline
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Introduction of the four conic sections, along with the degenerate conics
Definition of a circle
Derivation of the standard equation of a circle
Graphing circles
Solving situational problems involving circles
An Overview of Conic Sections
We introduce the conic sections (or conics), a particular class of curves which oftentimes appear
in nature and which have applications in other fields. One of the first shapes we learned, a circle,
is a conic. When you throw a ball, the trajectory it takes is a parabola. The orbit taken by each
planet around the sun is an ellipse. Properties of hyperbolas have been used in the design of
certain telescopes and navigation systems. We will discuss circles in this lesson, leaving
parabolas, ellipses, and hyperbolas for subsequent lessons.
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Circle (Figure 1.1) – when the plane is horizontal
Ellipse (Figure 1.1) – when the (tilted) plane intersects only one cone to form a bounded
curve
Parabola (Figure 1.2) – when the plane intersects only one cone to form an unbounded
curve
Hyperbola (Figure 1.3) – when the plane (not necessarily vertical) intersects both cones
to form two unbounded curves (each called a branch of the hyperbola)