Hyperbolas 10.3
Monday, May 12, 2014
5:30 PM
_______________________: the set of all points in a plane such that the absolute value of the
difference of the distances from two fixed points called the foci, is constant.
______________________: the midpoint of the segment connecting the foci (or vertices of a
hyperbola
______________________: a point on each branch of the hyperbola that is nearest the center
______________________: as a hyperbola recedes from the center, the branches approach these
lines
Transverse
axis
______________________:
segment of length 2a whose endpoints are vertices of the hyperbola
______________________: segment of the length 2b that is perpendicular to the transversal axis at
the center
Equations of Hyperbolas
Standard form of equation
Direction of Transverse Axis
Foci
Vertices
length of transverse axis (the abs. value of
the differences from any point on the hyp to
the foci)
length of conjugate axis
equation of asymptotes
To draw the graph:
1) draw the asymptotes
2) the point of intersection of the diagonals is the center of the hyperbola
 Ex. 1 Write an equation of a hyperbola with foci at (0,7) and (0,-7) if the length of the transverse axis
is 6 units.Graph
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 Ex. 2 Graph
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 Ex. 3 Determine the standard form of the equation, and determine the coordinates of the vertices
and foci. Graph. 144y2 - 25x2 -576y - 150x = 3249
4x2 - 9y2 - 32x - 18y + 19 = 0
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 Ex. 4 Find the standard form of the equation of the hyperbola with vertices at (3,2) and (3,-2) and
foci at (3,-5) and (3,5).
 Ex. 5 Find the standard form of the equations of the hyperbola having vertices at (3,-5) and (3,1) and
with the asymptotes: y = 2x - 8 and y = -2x + 4
 Ex. 6 The LORAN navigational system is based on hyperbolas. Two stations send out signals at the
same time. A ship notes the difference in the times at which it receives the signals. The ship is on a
hyperbola with the stations at the foci. Suppose a ship determines that the difference of its distance
from two stations is 50 nautical miles. The stations are 100 nautical miles apart. Write an equation
for a hyperbola on which the ship lies if the stations are (-50,0) and (50,0).
 Eccentricity of a hyperbola: (because c > a it follows that e >1)
 If the eccentricity is large the branches are nearly flat.
 If it is close to 1 they are more pointed.
The Equations of Conics:
The graph of Ax2 + Cy2 + Dx + Ey + F = 0
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1. Circle
2. Parabola
3. Ellipse
4. Hyperbola
Ex. Classify the equations
a. 3x2 - 2y2 + 4y - 3 = 0
b. 2y2 - 3x + 2 = 0
c. x2 + 4y2 - 3x - 3 = 0
d. x2 - 2x + 4y - 1 = 0
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