1. Hyperbolas

2. The standard form with center (h,k) is
Note: a is under the positive term. It is not necessarily
true that a is bigger than b.

3. Let’s take a look at the first hyperbola form.
V(h-a,k) c
V(h+a,k) b b
C(h,k)
F(h-c,k)
F(h+c,k) a
c is the distance
from the center
to the foci.
Note: If c is
the distance
from the center
to F, and all radii
of a circle = ,
then the hyp. of the right triangle is also c.
Therefore, to find c,
a2 + b2 = c2

4. Sketch the hyperbola whose equation is 4x2 - y2 = 16.
First divide by 16.
Write down a, b, c and the center pt.
a = 2
b = 4
Note: a is always under
the (+) term.
C(0,0)
Now find c.
Let’s sketch the hyperbola.

5. F F V V
Now, we need to find
the equations of the
asymptotes.
What are their slopes
and one point that is on both lines?

6. Sketch the graph of
4x2 - 3y2 + 8x +16 = 0
4(x2 + 2x ) - 3y2 = -16 +1
+ 4
4(x + 1)2 - 3y2 = -12
Now, divide by -12 and switch
the x and y terms.
C( , )
a =
b =
c =
e =
Sketch

7. V( , ) ( , )
F( , ) ( , ) F V V
Eq. of asymptotes. F

8. Classifying a conic from its general equation.
Ax2 + Cy2 + Dx + Ey + F = 0
If: A = C
AC = 0 , A = 0 or C = 0, but not both
AC > 0,
AC < 0
Both A and C = 0