Introduction to Conics
&
Circles
Chapter 11
Conics
The conics get their name from the fact that they
can be formed by passing a plane through a
double-napped cone (two right circular cones
placed together, nose-to-nose).
Conics
Conic sections were studied by the ancient
Greeks from a geometric point of view, but
today we describe them in terms of the
coordinate plane and distance, or as graphs
of equations.
Analytic Geometry
The study of the
geometric properties
of objects using a
coordinate system is
called analytic
geometry
(hence, the title of
chapter 11).
Typical Conic Shapes
Horizontal Parabola
Vertical Parabola
Circle
Vertical Ellipse
Horizontal Hyperbola
Vertical Hyperbola
First conic section:
CIRCLES
Definition of Circle
A circle is the set of all points that are the
same distance, r, from a fixed point (h, k).
Thus, the standard equation of a circle has
been derived from the distance formula.
Derive the equation for a circle
Given the distance formula, derive the standard
equation for a circle.
2
d=
𝑥2 − 𝑥1
+ 𝑦2 − 𝑦1
d=
𝑥−ℎ
2
+ 𝑦 −𝑘
2
r=
𝑥−ℎ
2
+ 𝑦 −𝑘
2
2
Standard Form of the Circle
(h, k) represents the __________
r represents the ___________
Example #1
Write an equation of a circle in standard form with a
center of (4, 3) and a radius of 5. Then graph the circle.
Example #2
Write an equation of a circle in standard form with a
center of (2, -1) and a radius of 4. Then graph the circle.
Example #3
Write the equation in standard form for the
circle centered at (–5, 12) and passing through
the point (–2, 8).
(x + 5)2 + (y – 12)2 = 25
General Form of the Circle
x2 + y2 + Ax + By + C = 0
Example #4
What is the equation of the circle pictured below?
Write the equation in both standard form and general form.
Example #5
Graph the circle.
x2 + y2 - 6x + 4y + 9 = 0