focus

Examples of focus in the following topics:

• Eccentricity

  • Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix.
  • In the next figure, each type of conic section is graphed with a focus and directrix.
  • The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix.
  • In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix.
  • This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus.

• Conics in Polar Coordinates

  • Previously, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line).
  • We can define any conic in the polar coordinate system in terms of a fixed point, the focus $P(r,θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.
  • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.  

• What Are Conic Sections?

  • For example, each type has at least one focus and directrix.
  • A focus is a point about which the conic section is constructed.
  • As with the focus, a parabola has one directrix, while ellipses and hyperbolas have two.
  • The point halfway between the focus and the directrix is called the vertex of the parabola.
  • The vertex lies at the midpoint between the directrix and the focus.

• Parabolas As Conic Sections

  • The point on the axis of symmetry where the right angle is located is called the focus.
  • The directrix is a straight line on the opposite side of the parabolic curve from the focus.
  • The parabolic curve itself is the set of all points that are equidistant (equal distances) from both the directrix line and the focus.
  • Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola traveling parallel to its axis of symmetry is reflected to its focus.
  • Conversely, light that originates from a point source at the focus is reflected, or collimated, into a parallel beam.

• Ellipses

  • The sun is at one focus of the ellipse (not at the center).
  • Similarly, the moon travels in an ellipse, with the Earth at one focus.
  • Each of these two points is called a focus of the ellipse.

• Reflections

  • For this section we will focus on the two axes and the line $y=x$.

• Standard Equations of Hyperbolas

  • The eccentricity e equals the ratio of the distances from a point P on the hyperbola to one focus and its corresponding directrix line (shown in green).

• Factors

  • For now, we will focus on factoring whole numbers.

• Introduction to Systems of Equations

  • In this section, we will focus primarily on systems of linear equations which consist of two equations that contain two different variables.

• Applications of the Parabola

  • Conversely, a parabolic reflector can collimate light from a point source at the focus into a parallel beam.