Chapter 7 Analyzing Conic Sections

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Chapter 7Analyzing Conic Sections
Jennifer Huss
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7-1 The Distance and Midpoint Formulas

• To find the distance between any two points
  (a, b) and (c, d), use the distance formula
• Distance (c a)2 (d b)2
• The midpoint of a line is halfway between the two
  endpoints of a line
• To find the midpoint between (a, b) and (c, d),
  use the midpoint formula
• Midpoint (a c) , (b d)
• 2 2

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7-1 Example

• Find the distance between (-4, 2) and (-8, 4).
  Then find the midpoint between the points.

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7-1 Problems

• Find the distance between (0, 1) and (1, 5).
• Find the midpoint between (6, -5) and (-2, -7).
• Find the value for x if the Distance 53 and
  the endpoints are (-3, 2) and (-10, x).
• If you are given an endpoint (3, 2) and midpoint
  (-1, 5), what are the coordinates of the other
  endpoint?

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7-2 Parabolas

• A parabola is a set of points on a plane that are
  the same distance from a given point called the
  focus and a given line called the directrix
• The axis of symmetry is perpendicular to the
  directrix and passes through the parabola at a
  point called the vertex
• The latus rectum goes through the focus and is
  perpendicular to the axis of symmetry
• If the equation of the parabola begins with x
  then the parabola is not a function (fails the
  vertical line test)

Axis of Symmetry
Parabola
Focus
Latus Rectum
Vertex
Directrix
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7-2 Parabolas (cont.)
Important Information About the Parabolas
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7-2 Example

• Write y x2 4x 1 in the form y a (x h)2
  k and name the vertex, axis of symmetry, and
  the direction the parabola opens.

You can always check your answers by graphing.
y x2 4x 1 y (x2 4x o ) 1 o y
(x2 4x 4) 1 4 y (x 2)2 3
Vertex (-2, -3) Axis of Symmetry x -2 The
parabola opens up because a 1 so a gt 0.
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7-2 Problems

• Graph the equation x2 8y.
• For the parabola y2 -16x name the vertex,
  focus, length of latus rectum, and direction of
  opening. Also, give the equations of the
  directrix and axis of symmetry.
• Given the vertex (4, 1) and a point on the
  parabola (8, 3), find the equation of the
  parabola.

Graph for 1
Graph for 2
2) Vertex (0,0) Focus (-4,0) Latus rectum 16
Direction left Directrix x 4 Axis of
symmetry y 0 3) y (1/8)(x 4)2 1
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7-3 Circles

• A circle is a set of points equidistant from a
  center point
• The radius is a line between the center and any
  point on the circle
• The equation of a circle is (x h)2 (y
  k)2 r2 where the radius is r and the vertex is
  (h, k)
• Sometimes you need to complete the square twice
  to get the equation in this form (once for x and
  once for y)

Radius (r)
Vertex (k, h)
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7-3 Examples

• Find the center and radius of x2 y2 4x 12y
  9 0 and then graph the circle.

x2 4x o y2 12y o 9 o o x2 4x
4 y2 12y 36 9 4 36 (x 2)2 (y
6)2 49 Radius 7 and Center is (-2, 6)
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7-3 Examples (cont.)

• If a circle has a center (3, -2) and a point on
  the circle (7, 1), write the equation of the
  circle.

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7-3 Problems

• Find the center and radius of x2 y2 4y 0.
  Then graph the circle.
• If a circle has a center (0, 0) and a point on
  the circle (-2, -4) write the equation of the
  circle.

1) x2 (y 2)2 4 Center (0, -2) and Radius
2 2) x2 y2 20
1
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7-4 Ellipses

• An ellipse is the set of all points in a plane
  such that the sum of the distances from the foci
  is constant
• An ellipse has two axes of symmetry
• The axis of the longer side of the ellipse is
  called the major axis and the axis of the shorter
  side is the minor axis
• The focus points always lie on the major axis
• The intersection of the two axes is the center of
  the ellipse

Major Axis
Focus
Center
Minor Axis
Focus
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7-4 Ellipses (cont.)
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1

• Important Notes
• In the above chart, c a2 b2
• a2 gt b2 always so a2 is always the larger
  number
• If the a2 is under the x term, the ellipse is
  horizontal, if the a2 is under the y term the
  ellipse is vertical
• You can tell that you are looking at an ellipse
  because x2 is added to y2 and the x2 and y2
  are divided by different numbers (if numbers were
  the same, its a circle)

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7-4 Example

• Given an equation of an ellipse 16y2 9x2 96y
  90x -225 find the coordinates of the center
  and foci as well as the lengths of the major and
  minor axis. Then draw the graph.

Center (5, 3) 16 gt 9 so the foci are on the
vertical axis c 16 9 c 7 Foci ( 5
7, 3) and (5 7, 3) Major Axis Length 4 (2)
8 Minor Axis Length 3 (2) 6
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7-4 Problems

• For 49x2 16y2 784 find the center, the foci,
  and the lengths of the major and minor axes.
  Then draw the graph.
• Write an equation for an ellipse with foci (4, 0)
  and (-4, 0). The endpoints of the minor
  axis are (0, 2) and (0, -2).

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7-5 Hyperbolas

• A hyperbola is a set of all points on a plane
  such that the absolute value of the difference
  (subtraction) of the distances from a point to
  the two foci is constant
• The center is the midpoint of the segment
  connecting the foci
• The vertex is the point on the hyperbola closest
  to the center
• The asymptotes are lines the hyperbola can
  approach but never touch
• The transverse axis goes through the foci
• The conjugate axis is perpendicular to the
  transverse axis at the center point

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7-5 Hyperbolas (cont.)
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You must be looking at a hyperbola because the x2
and y2 terms are subtracted (x2 y2) or
(y2 x2)
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7-5 Example

• Write the standard form of the equation of the
  hyperbola 144y2 25x2 576y 150x 3249.
  Then find the coordinates of the center, the
  vertices, the foci, and the equation of the
  asymptotes. Graph the hyperbola and the
  asymptotes.

144(y2 4y o) 25(x2 6x o) 3249
144(o) 25(o) 144(y2 4y 4) 25(x2 6x 9)
3249 144(4) 25(9) 144(y 2)2 25(x 3)2
3600 (y-2)2 _ (x 3)2 25 144
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Center (-3, 2) a 5 so the vertices are
(-3, 7) and (-3, -3) a2 b2 c2 25 144 c2 c
13 The foci are (-3, 15) and (-3, -11).
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7-5 Example (cont.)

• Asymptotes have the formula y /- a/b x and we
  have center (-3, 2) and slopes /- 5/12.
• y 2 5/12 (x 3) y 2 -5/12 (x 3)
• y 2 (5/12) x 15/12 y 2 (-5/12) x
  -15/12
• y (5/12) x 13/4 y (-5/12) x 3/4

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7-5 Problems

• Find the coordinates of the vertices and the
  foci. Give the asymptote slopes for each
  hyperbola. Then draw the graph.
• x2 _ y2
• 9 49
• 2) 25x2 4y2 100

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1)
2)
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7-6 Conic Sections

• Circles, ellipses, parabolas, and hyperbolas are
  all formed when a double cone is sliced by a
  plane
• The general equation of any conic section is
  Ax2 Bxy Cy2
  Dx Ey F 0
• The standard equations for each specific conic
  section are listed in previous sections
• If B 0 and you look at A and C in the
  equations

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7-6 Example

• Identify 9x2 16y2 54x 64y 1 0 as one
  of the four conic sections. Then graph the conic
  section.

• 9x2 16y2 54x 64y -1
• 9 (x2 6x o) 16(y2 4y o) -1 9(o)
  16(o)
• 9 (x2 6x 9) 16(y2 4y 4) -1 9(9)
  16(4)
• 9(x 3)2 16(y 2)2 144
• (x 3)2 (y 2)2
• 9
• This conic section is an ellipse.

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7-6 Problems

• Write the equation in standard form and decide if
  the conic section is a parabola, a circle, an
  ellipse, or a hyperbola. Then graph the
  equation.
• x2 y2 6x 7
• 5x2 6y2 30x 12y 9 0

1)
2)
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7-7 Solving Quadratic Systems

• When you solve a system of quadratic equations
  the method is almost the same as solving a system
  of linear equations
• If the system has one equation of a conic section
  and one equation of a straight line, you can get
  zero, one, or two solutions to the system
• If both the equations are conic sections, the
  system should have zero, one, two, three, or four
  solutions

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7-7 Example

• Solve this system of equations using algebraic
  methods and by graphing the equations.

The system of equations has one solution, (0, 5).
The graphs of these equations confirms this.
Set the equations equal to each other to solve
for x. -4x 5 (x 2)2 1 -4x 5 x2 4x
4 1 -4x 5 x2 4x 5 5 x2 5 x2 0 x
0
Then put x 0 back in to solve for y. y -4(0)
5 y 5
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7-7 Problems

• Solve these systems of equations by using algebra
  and graphing the equations.
• 4x2 y2 25 2) x2 y2 10
• 2x2 y2 -1 y x2 4