An Introduction
Conic Sections - IntroductionA conic is a shape
generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant.
Conic Sections - IntroductionThe resulting
collection of points is called a right circular cone. The two parts of the cone intersecting at the vertex are called nappes.
Vertex
Nappe
Conic Sections - IntroductionA “conic” or conic
section is the intersection of a plane with the cone.
The plane can intersect the cone at the vertex resulting in a point.
Conic Sections - IntroductionThe plane can
intersect the cone perpendicular to the axis resulting in a circle.
Conic Sections - IntroductionThe plane can
intersect one nappe of the cone at an angle to the axis resulting in an ellipse.
Conic Sections - IntroductionThe plane can
intersect one nappe of the cone at an angle to the axis resulting in a parabola.
Conic Sections - IntroductionThe plane can intersect
two nappes of the cone resulting in a hyperbola.
Graph each equation. Describe the graph, find the lines of symmetry, x and y intercepts, domain, and range.
1.
2.
3.
1622 yx
3649 22 yx
422 yx
162.3
81.2
100425.1
22
22
22
yx
yx
yx
What is a circle?A circle is the set
of points equally distant from one central point.
The central point is called the center.
••
Center
What does r represent?The distance from the center to the curve of
the circle is called the radius.
r
What does d represent?The diameter is the distance across the
circle.
d
Assume that (x,y) are the coordinates of a point on the circle.
Use the distance formula to find the radius.
•
•
(x,y)
(h,k)
Equation of a circler2=(x - h)2 + (y – k)2
Let’s investigate!
Example #1Find the equation of a circle whose center is
at (2, -4) and the radius is 5.
Let’s check our answer.
Example #2Write an equation of a circle if the endpoints
of a diameter are at (5,4) and (-2, -6).Hint: Draw a picture, then find the
center and radius.
Example #3Find the center and radius of the circle with equation x2 + y2 = 25.
Graph the circle.
Example #4Find the center and radius of the circle with equation
x2 + y2 – 4x + 8y – 5 = 0.
On your own…
1. Find center and radius and graph: x2 + y 2 -10x +8y = -40
2. Write an equation for a circle that passes through the point (-1, 4) with a center at (-3, 6).
Assignment:10.1 p. 550 #1, 3, 5, 17-28, 62
10.3 p. 564 #27-31 odd, 35, 43, 45, 47, 49, 61, 63, 73, (78 graph)