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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.

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conics
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).

conics1
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
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
Typical Conic Shapes

Horizontal Parabola

Circle

Horizontal Hyperbola

Vertical Parabola

Vertical Ellipse

Vertical Hyperbola

definition of circle
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
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
Standard Form of the Circle

(h, k) represents the __________

r represents the ___________

example 1
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
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
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
General Form of the Circle

x2+ y2 + Ax + By + C = 0

example 4
Example #4

What is the equation of the circle pictured below?

Write the equation in both standard form andgeneral form.

example 5
Example #5

Graphthecircle.

x2+ y2 - 6x + 4y + 9 = 0