Introduction to Conics&

Circles

Chapter 11

ConicsThe 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).

ConicsConic 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 Circle

Vertical Parabola 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.

d =

d =

r =

Standard Form of the Circle(h, k) represents the __________r represents the ___________

Example #1Write an equation of a circle in standard form with a

center of (4, 3) and a radius of 5. Then graph the circle.

Example #2Write 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 #4What 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