Stewart Calculus Section 10.5

1. 10.5 Conic Sections
Three types of conic sections:
Parabola
Ellipse
Hyperbola

2. A parabola is the set of points in a plane
that are equidistant from a fixed point
(called the focus) and a fixed line (called
the directrix). The point half way between the
focus and the directrix is called the vertex.
The line through the focus perpendicular to
the directrix is called the axis of the
parabola.
axis
focus
vertex
directrix

3. An equation of the parabola with focus ( , )
and directrix = is = .
An equation of the parabola with focus ( , )
and directrix = is = .
y
=
( , )
x
=

4. An ellipse is the set of points in a plane that
the sum of whose distances from two fixed
points (called the foci) is a constant. The
line through the foci is called the major axis
of the ellipse. The intersecting points of the
major axis and the ellipse are called the
vertices.
axis
vertices foci

5. The ellipse + = >
has foci (± , ) where = , and
vertices (± , ).
The ellipse + = >
has foci ( , ± ) where = , and
vertices ( , ± ).
y
( , ) + =
( , ) ( , )
( , ) ( , ) x
( , )

6. An hyperbola is the set of points in a plane
that the difference of whose distances from
two fixed points (called the foci) is a
constant. The line through the foci is called
the axis of the ellipse. The intersecting points
of the major axis and the ellipse are called
the vertices. Both branches of the hyperbola
approach the asymptotes.
foci
axis
vertices

7. The hyperbola = has foci (± , )
where = + , vertices (± , ) , and
asymptotes = ±( / ) .
The hyperbola = has foci ( ,± )
where = + , vertices ( , ± ) , and
asymptotes = ±( / ) .
y
( , )
( , ) x

8. Ex: What can you say about the conic?
+ + =

9. Ex: What can you say about the conic?
+ + =
Complete the square:
( ) ( )+ =

10. Ex: What can you say about the conic?
+ + =
Complete the square:
( ) ( )+ =
( + ) ( + )+ =

11. Ex: What can you say about the conic?
+ + =
Complete the square:
( ) ( )+ =
( + ) ( + )+ =
( ) ( ) + =

12. Ex: What can you say about the conic?
+ + =
Complete the square:
( ) ( )+ =
( + ) ( + )+ =
( ) ( ) + =
( ) ( )
=

13. Ex: What can you say about the conic?
+ + =
Complete the square:
( ) ( )+ =
( + ) ( + )+ =
( ) ( ) + =
( ) ( )
=
It is a hyperbola, = , = , = + = .
The foci are ( , ± ). The vertices are ( , )
and ( , ). The asymptotes are − = ± ( − ).