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9.2 – Geometry of Conic Sections

Key Terms

  • Asymptote – A line on a graph that a curve approaches more and more closely without intersecting.
  • Center of a Hyperbola – The point halfway between the vertices of a hyperbola, or the midpoint of the transverse axis of a hyperbola.
    • The center of a hyperbola is the point where the asymptotes intersect.
  • Center of an Ellipse – The midpoint of a line segment joining the foci of an ellipse.
    • The center of an ellipse is the point where the major axis and minor axis intersect.
  • Directrix – A line that can be used to define a parabola. The distance from any point on the parabola to the focus is the same as the distance from that point to the directrix.
  • Focus of a Hyperbola – One of the two points that can be used to define a hyperbola. For every point on a hyperbola, the distance from the point to one focus, minus the distance from the point to the other focus, is equal to some constant value.
    • The plural of focus is foci.
    • Another name for a focus is a focal point.
  • Focus of a Parabola – A point that can be used to define a parabola.
    • The distance from any point on the parabola to the focus is the same as the distance from that point to a line called the directrix.
    • Another name for the focus is the focal point.
  • Focus of an Ellipse – One of the two points that can be used to define an ellipse.
    • For every point on an ellipse, the distance from the point to one focus, plus the distance from the point to the other focus, is equal to some constant value.
    • Another name for a focus is a focal point.
    • The plural of focus is foci.
  • Major Axis – The longest line segment through the center of an ellipse that has its endpoints on the ellipse.
  • Minor Axis – The shortest line segment through the center of an ellipse that has its endpoints on the ellipse.
  • Transverse Axis – The line segment that joins the two vertices of a hyperbola.
    • The length of the transverse axis is equal to the difference between the distance from any point on the hyperbola to one focus, and the distance from that point to the other focus.
  • Vertex of a Hyperbola – The point on each branch of a hyperbola that is closest to the other branch of the hyperbola.
    • The plural of vertex is vertices.
    • The vertices are the endpoints of the transverse axis, which is a line of symmetry for a hyperbola.
  • Vertex of a Parabola – The point on a parabola where it changes directions.
    • The vertex is the point where the parabola intersects its axis of symmetry.

Review

 Conic Sections in Geometry
  • Conic sections are made by intersecting a cone with a plane, so it makes sense that geometry would play a role in conic sections.
    • Ex. Imagine a flashlight beam scanning the ground.
    • The light itself forms a cone.
    • You can think of the flashlight lens and the ground as planes.
    • The way in which this light intersects with the lens of the flashlight and also with the ground creates conic sections.

GeoB 9.2 Flashlight

Plurals
  • The plural of axis is axes.
  • The plural of focus is foci.
  • The plural of vertex is vertices.
 Distance
  •  Distances are always positive.
    • When you subtract two distances, use their absolute values to find the positive distance.
    • Remember: big # minus small #.

Notes

 Conic Sections: Circles
  • Circle – The set of all points in a plane that are a certain distance from a single point.
  • Radius – The distance from the center to any point on the circle.

GeoB 9.2 Circle

 Conic Sections: Ellipses
  • Ellipse – The set of all points in a plane for which the sum of the distances to two fixed points equals a certain constant.
    • In the ellipse shown below, each point marked with a dot is called a focus.
    • The foci of an ellipse will always lie inside the ellipse.
    • The foci of an ellipse lie on the major axis of the ellipse.

GeoB 9.2 Focus


 

GeoB 9.2 Ellipse Parts GeoB 9.2 Ellipse

  • How to Find the Constant Value of an Ellipse
    1. Choose any point on the ellipse.
    2. Measure the distances from that point to each focus.
    3. Add the two measures
  • In the ellipse shown below, the red line segment is called the major axis.

GeoB 9.2 Major Axis

  • In the ellipse shown below, the red line segment is called the minor axis.

GeoB 9.2 Minor Axis

  • How are circles like ellipses?
    • All circles are ellipses, but only some ellipses are circles.

GeoB 9.2 Circles are Ellipses

GeoB 9.2 Circle vs Ellipse

 Conic Sections: Hyperbolas
  • A hyperbola
    • has 2 foci.
    • The vertex is the extreme point on half of a hyperbola.
    • The center is the point halfway between its two vertices.
    • The line segment that connects the two vertices of a hyperbola is called the transverse axis.
    • The length of the transverse axis is equal to the difference between the distances from any point on the hyperbola to each focus.

GeoB 9.2 Hyperbola Parts GeoB 9.2 Hyperbola

  • How to Find the Constant Value of a Hyperbola
    1. Choose any point on the hyperbola.
    2. Measure the distances from that point to each focus.
    3. Subtract the two measures
  • Asymptotes – Two lines that intersect at the center of a hyperbola.
    • Moving from the center, the two curves of the hyperbola get closer and closer to the asymptotes, but they never touch them.
    • Think of them like electric fences.  You cannot touch them, ever!

GeoB 9.2 Asymptote

 Conic Sections: Parabolas
GeoB 9.2 Parabola Parts GeoB 9.2 Parabola Shapes GeoB 9.2 Parabola
  • The vertex of a parabola
    • Is the extreme point of a parabola and is located halfway between the focus and directrix.
    • It’s the point where the parabola makes its sharpest turn.
  • Symmetry
    • A parabola has one line of symmetry.
    • The line goes through the parabola’s focus and is perpendicular to its directrix.

GeoB 9.2 Parabola Symmetry

 

Comparing Conic Sections
GeoB 9.2 Chart GeoB 9.2 Circles to Hyperbolas

Examples

  • Ex 1. Both the red and blue line segments stretch from the center of the circle to a point on the circle. The length of the blue line segment is 8. How long is the red line segment?
    • Answer: 8

GeoB 9.2 Q1-6

  • Ex 2. The length of the red line segment is 8, and the length of the blue line segment is 3. How long is the major axis of the ellipse?
    • Answer: 11

GeoB 9.2 Q1-7

  • Ex 3. The length of the major axis of the ellipse below is 17, and the length of the red line segment is 6. How long is the blue line segment?
    • Answer: 11

GeoB 9.2 Q1-8

  • Ex 4. The length of the major axis of the ellipse below is 13. What is the sum of the lengths of the red and blue line segments?
    • Answer: 13

GeoB 9.2 Q1-9

  • Ex 5. The length of the red line segment is 18, and the length of the blue line segment is 8. How long is the transverse axis?
    • Answer: 10

GeoB 9.2 Q2-1

  • Ex 6. The distance from the green point on the parabola to the parabola’s focus is 7. What is the distance from the green point to the directrix?
    • Answer: 7

GeoB 9.2 Q2-3

  • Ex 7. Look at the hyperbola graphed below. The hyperbola gets very close to the red lines on the graph, but it never touches them. Which term describes each of the red lines?
    • Answer: Asymptote

GeoB 9.2 Q2-10


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