Introduction to Conic Sections
A conic section is a curve formed
by the intersection of
_________________________
a plane and a double cone.
Circles
The set of all points that are the same
distance from the center.
Standard Equation:
( x  h)  ( y  k )  r
2
With CENTER: (h, k)
& RADIUS: r (square root)
2
2
(h , k)
r
Ellipse
Basically an ellipse is a squished circle
( x  h)
( y  k)

1
2
2
a
b
2
Standard Equation:
(h , k)
a
2
b
Center: (h , k)
a: major radius (horizontal), length from center to edge of circle
b: minor radius (vertical), length from center to top/bottom of circle
* You must square root the denominator
The Ellipse
• Tilt a glass of water
and the surface of
the liquid acquires
an elliptical outline.
• Salami is often cut
obliquely to obtain
elliptical slices
which are larger.
• On a far smaller
scale, the electrons of
an atom move in an
approximately
elliptical orbit with the
nucleus at one focus.
• Any cylinder sliced
on an angle will
reveal an ellipse in
cross-section
• (as seen in the
Tycho Brahe
Planetarium in
Copenhagen).
Example
This must
equal 1
( x  4)
( y  5)

1
25
4
2
a²
2
b
Center: (-4 , 5)
a: 5
b: 2
2
Parabola
vertex
We’ve talked about this before…
a U-shaped graph
vertex
Standard Equations:
( x  h) 2  4 p ( y  k )
This equation opens up or down
OR
( y  k ) 2  4 p ( x  h)
This equation opens left or right
HOW DO YOU TELL…LOOK FOR THE SQUARED VARIABLE
Vertex: (h , k)
•If there is a negative in front of the squared variable, then it opens down or left.
•If there is NOT a negative, then it opens up or right.
• The easiest way to
visualize the path of a
projectile is to
observe a waterspout.
• Each molecule of
water follows the
same path and,
therefore, reveals a
picture of the curve.
Hyperbolas
What I look like…two parabolas, back to back.
Standard Equations:
( x  h)
( y  k)

1
2
2
a
b
2
2
This equation opens left and right
OR
( y  k)
( x  h)

1
2
2
a
b
2
2
This equation opens up and down
Have I seen this before? Sort of…only now we have a minus sign in the middle
Center: (h , k)
(h , k)
x  y 1
2
2
1
2
x  1  ( y  2)
12
( x  2)  y  4
2
2