introduction to conics & circles chapter 11
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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. - PowerPoint PPT PresentationTRANSCRIPT
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