1. Conic Sections

2. Circles Ellipses Parabolas Hyperbolas Systems Day 1 Day 2 Day 1 Day 1

3. Circles
Definitions
1) A circle is the set of all points, in a plane, equidistant from a fixed point.
2) A circle is the intersection of a right circular cone and a plane perpendicular to the axis of the cone.
General form:
where
Standard form:
Center:
(h, k) r
Radius: r

4. Find the center, radius
and graph.
Center:
r =
Center:
r =
Center:
r =
Center:
r =

5. Write the equation of the circle
The diameter of the circle has endpoints at (2, 6) and (8, -2)
The center is at (2, -4) and the circle is tangent to the x-axis
The center is at (4, -2) and the circle passes through the point (5, 3)
Center: (-3, 0) and r =
Center: (2, -1) and r = 8

6. Write in standard form by completing the square

7. Ellipses
Definitions
An ellipse is the set of all points, in a plane, such that the sum of the distances of each point from two fixed points is a constant
2) An ellipse is the intersection of a right circular cone and a plane not perpendicular to the axis of the cone. F C
Vocabulary:
Foci - Each fixed point is called a focus of the ellipse. F
Center: the midpoint of the line segment joining the foci
and the ellipse
Major Axis: a line segment with endpoints on the ellipse and containing the foci
Minor Axis: line segment with endpoints
on the ellipse and perpendicular to the
major axis at the center of the ellipse F F C

8. The major axis is horizontal 2a The major axis is vertical
General form:
where
Horizontal
Vertical
Standard Equation
Center
Length of Major Axis
Length of minor axis
How to find c
Length of Focal Chord
Foci
located at
(h, k)
(h, k) 2a
The major axis is horizontal 2a
The major axis is vertical 2b
The minor axis is vertical 2b
The minor axis is horizontal 2c 2c
(h - c, k) and (h + c, k)
(h, k - c) and (h, k + c)

9. Find the center, foci, length of major and minor axes and graph.
Length of Major axis
Length of minor axis

10. Write an equation for the graph.

11. Center:
Foci:
Length of Major axis
Length of minor axis

12. Ellipses
Write in standard form by completing the square

13. Write the equation of the ellipse in standard form.
The foci are at (0,-3) and (0,3). The length of the minor axis is 4.
The major axis is 16 units long and parallel to the x-axis. The center is at (5, 4) and minor axis is 9 units long.
The vertices are at (-11, 5) and (7,5) and the minor axis is 4 units long.

14. Parabolas Definitions
A parabola is the intersection of right circular cone and a plane parallel to an element of the cone
2) A parabola is the set of points in a plane each of which is the same distance from a fixed point as it is from a fixed line.
Vocabulary: F
Focus - the fixed point (always located inside the parabola) V
Directrix - the fixed line (never intersects the parabola)
Vertex - the point at which the axis intersects the parabola
Axis of Symmetry - a line drawn through the focus, perpendicular to the
directrix (sometimes called the axis of the parabola)
Focal Chord - any segment joining two points on the parabola
and passing through the focus
Focal Radius - any segment joining the focus to a point on the
parabola V F
Latus Rectum - the focal chord which is perpendicular to the
axis and contains the focus.

15. General form: Where but not both (h, k) (h, k) y = k x = h (h + c, k)
Horizontal
Vertical
Standard Equation
Direction of opening
Vertex
Axis of
Symmetry
Location of focus
Directrix
Length of
Latus Rectum
If > 0 opens up
If > 0 opens right
If < 0 opens left
If < 0 opens down
(h, k)
(h, k)
y = k
x = h
(h + c, k)
(h, k + c)
x = h - c
y = k - c
|4c|
|4c|

16. Graph then find the requested information.
Vertex:
Axis of symmetry
Focus:
Equation of directrix
(Length of latus rectum = )
Endpoints of latus rectum:

17. Write the equation of the parabola in standard form.

18. Write the equation of the parabola in standard form.
The vertex is (-7, 4), the length of the latus rectum is 6 and the graph opens left.
The focus is at (3, 8) and the directrix is y = 4.
The directrix is y = 2 and the right endpoint of the latus rectum is (6, -2)

19. Hyperbolas Definitions
1) An hyperbola is a set of points in a plane such that for each, the absolute value of the difference of its distances from the two fixed points is a constant.
2) An hyperbola is the intersection of a right circular cone and a plane cutting both nappes of the cone.
Vocabulary
Foci - the two fixed points
Center - the midpoint of the transverse axis
Transverse Axis - a segment of the line passing through the foci with
vertices as endpoints
Vertices - points of intersection of the branches of the hyperbola and
the transverse axis
Conjugate Axis - perpendicular to the transverse axis at the center
Asymptote - a straight line approached by a given curve as one of
the variables in the equation of the curve approaches infinity

20. General form: Where (h, k) (h, k) (h, k – a) and (h, k + a)
Horizontal
Vertical
Standard Equation
Center
Vertices
Length of Transverse Axis
Length of Conjugate Axis
How to find c
Foci
located at
Equation of Asymptotes
(h, k)
(h, k)
(h, k – a) and (h, k + a)
(h – a, k) and (h + a, k) 2a 2a 2b 2b
c2 = a2 + b2
c2 = a2 + b2
(h – c, k) and (h + c, k)
(h, k - c) and (h, k + c)

21. Graph the hyperbola and find the requested information.
Center:
Vertices:
Foci:
Equation of asymptotes:

22. Write an equation for the graph.

23. Write the equation of the hyperbola in standard form.

24. Write the equation of the hyperbola in standard form.
The center is (-3, 2), a vertex is (-3, 5) and one endpoint of the conjugate axis is -8, 2)
A vertex is (-4, 4) and the equation of the asymptotes is
A vertex is (4, 0) and the foci are at (6, 0) and (-6,0)

25. Identify the type of Conic section
Ellipse
Hyperbola
Parabola
Circle

26. Identify the type of conics, sketch a graph and solve the system.