Transcript standard form equation of a hyperbola

SECTION: 10 – 3 HYPERBOLAS
WARM-UP
Find the center, vertices, foci, and eccentricity of each
ellipse.
1.
 x  5
81
2
y  1


100
2
1
2. 16 x  9 y  32 x  72 y  16  0
2
2
3. 4 x  25 y  16 x  150 y  141  0
2
2
HYPERBOLA. A hyperbola is the set of all points
(x,y) the difference of whose distances (d1,d2) from
two distinct fixed points (foci) is a positive constant.
DESCRIPTION
BRANCHES. Every hyperbola has two disconnected
branches, which form the curve of the hyperbola.
TRANSVERSE AXIS. The transverse axis is the line
segment passing through the two foci and the two
vertices.
CENTER. The midpoint of the transverse axis is the
center of the hyperbola.
 x, y 
d1
FOCUS
TRANSVERSE
d2
c
a
CENTER
VERTICES
FOCUS
STANDARD FORM EQUATION OF A
HYPERBOLA. The standard form equation of a
hyperbola with center at (h,k) is:
1. When the
transverse
axis is horizontal:
2
2
 x  h   y  k   1
2
2
a
b
2. When the transverse axis is vertical:
2
2
 y  k    x  h  1
2
2
a
b
The vertices are a units from the center, and foci are c
units from the center, where c2=a2+b2.
WRITE THE STANDARD FORM OF THE
EQUATION OF A HYPERBOLA
1. Determine whether the transverse axis is oriented
horizontally or vertically.
2. Find the coordinates of the center of the
hyperbola.
3. Find the value of c, which is the distance between
the center and the foci.
4. Find the value of a, which is the distance between
the center and the vertices.
5. Find the value of b, using the formula
b  c2  a2 .
6. Substitute the values of a and b into the standard
form of the equation of the hyperbola.
EXAMPLE 1. Find the standard form of the equation
of the hyperbola with foci at (–1,2) and (5,2) and
vertices at (0,2) and (4,2).
ASYMPTOTES OF A HYPERBOLA. Each
hyperbola has two slant asymptotes that intersect at
the center of the hyperbola.
1. When the transverse axis is oriented horizontally,
the asymptotes are:
b
y  k   x  h
a
2. When the transverse axis if oriented vertically,
the asymptotes are:
a
y  k   x  h
b
EXAMPLE 2. Find the equations of the slant
2
2
4
x

3
y
 8 x  16  0.
asymptotes of the hyperbola
USING ASYMPTOTES TO FIND THE
STANDARD EQUATION OF A HYPERBOLA
1. Determine the center of the hyperbola by finding
the point of intersection of the asymptotes. Use
any method to solve the system of linear
equations.
2. Determine the values of a and b. Let m1 be the
b
positive slope, then m1  (horizontal) or
a
a
m1  (vertical).
b
3. Substitute the values of h, k, a, and b into the
standard form of the equation.
EXAMPLE 3. Determine the standard form of the
equation of the hyperbola with vertices (3,–5) and
(3,1) and asymptotes y=2x–8 and y=–2x+4.
SKETCHING THE GRAPH OF A HYPERBOLA
1. Determine whether the transverse axis is oriented
horizontally or vertically.
2. Determine the values of a, b, and c, if necessary.
3. Determine and plot the center of the hyperbola.
4. Determine and plot the vertices, of the hyperbola.
a. If the transverse axis is oriented horizontally,
the vertices are (h+a,k) and (h–a,k) .
b. If the transverse axis is oriented vertically, the
vertices are (h,k+a) and (h,k–a).
4. Determine the conjugate axes.
a. If the transverse axis is oriented horizontally,
the conjugate axes are (h,k+b) and (h,k–b).
b. If the transverse axis is oriented vertically, the
conjugate axes is (h+b,k) and (h–b,k).
5. Construct a box that passes through the both
vertices and the points marking the conjugate
axes.
6. Determine the equation of the slant asymptotes.
Then graph the slant asymptotes. The slant
asymptotes pass through opposite corners of the
box constructed in Step 5.
7. Draw the curve of the hyperbola, which passes
through both vertices and converges to the slant
asymptotes.
EXAMPLE 4. Sketch the hyperbola 4 x  y  16.
2
2
y
x
GENERAL FORM OF THE EQUATION OF A
CONIC SECTION. The general form of the equation
of a conic section is:
Ax 2  Cy 2  Dx  Ey  F  0
CLASSIFYING CONIC SECTIONS. To classify
the type of conic section, given the general form of
the equation, use the following rules.
1. If A=C, then the conic section is a circle.
2. If AC=0, then the conic section is a parabola.
3. If AC>0, then the conic section is an ellipse.
4. If AC<0, then the conic section is a hyperbola.
EXAMPLE 5. Classify each conic section.
a. 4 x  9 x  y  5  0
2
b. 4 x 2  y 2  8 x  6 y  4  0
c. 2 x  4 y  4 x  12 y  0
2
2
d. 2 x  2 y  8 x  2  0
2
2
CLASS WORK/HOMEWORK:
SECTION: 10 – 3
PAGE: 720 – 721
PROBLEMS: 7, 9, 11, 23, 25, 27, 29, 39, 44 – 51 All