The Conic Sections
Chapter 10
Introduction to Conic Sections (10.1)
A conic section is the intersection
of a plane with a double-napped
cone.
How are these sections created?
By changing the angle and the
location of the intersection, a
parabola, circle, ellipse or hyperbola
is produced.
Circle
(Section 10.2)
When a plane
intersects a
double-napped
cone and is
parallel to the
base of a cone, a
circle is formed.
Ellipse
(Section 10.3)
 When a plane
intersects a doublenapped cone and is
neither parallel nor
perpendicular to the
base of the cone, an
ellipse can be
formed. The figure
is a closed curve.
Hyperbola (Section 10.4)
 When a plane
intersects a doublenapped cone and is
neither parallel or
perpendicular to the
base of the cone, a
hyperbola can be
formed. The figure
consists of two open
curves.
Parabola (Section 10.5)
 When a plane
intersects a doublenapped cone and is
parallel to the side
of the cone, a
parabola is formed.
Foundational Knowledge – Chapter 10
The Distance Formula
d  ( x2  x1 )2  ( y2  y 1 )2
Example: find the distance between the points (4, -2) and (8, 3)
Answer is
41
Remember: if the answer is not a perfect square,
leave as a simplified radical expression.
Foundational Knowledge
•
The Midpoint Formula
 x1  x2 y1  y2 
m. p.  
,

2
2


(The midpoint is the average of the two coordinates!)
Example: Find the midpoint between (-2, 4) and (6, -5)
… it’s (2, -1/2)
•
The Slope Formula
m
y2  y1
x2  x1
Example: Find the slope of the two points listed above.
9
m
8
Putting it all together: the parallelogram example
Example:
Using a combination of these two tools, determine if the four
sided shape with vertices ( -2, 3) (-3, -2) (2, -3) (3,2) is a
parallelogram.
Remember: to prove a parallelogram, show that either…




One pair of opposite sides congruent (distance) and
parallel (slope) or,
Opposite sides are parallel (slope) or
Diagonals bisect each other (midpoint) or
Both pairs of opposite sides same length (distance)
( -2, 3) (-3, -2) (2, -3) (3,2)
m1 
2  3
5

5
3  ( 2) 1
Compare slope and
length of one pair of
opposite sides
m2 
2  ( 3) 5
 5
3 2
1
d1  ( 3  ( 2)) 2  ( 2  3) 2
d1  (3  2)2  (2  ( 3)) 2
d1  1  25  26
d1  1  25  26
Conclusion: shape is a parallelogram