focal point

Examples of focal point in the following topics:

• Introduction to Hyperbolas

  • The set of all points such that the difference between the distances to two focal points is constant
  • Then the difference of distances between $P$ and the two focal points is:
  • Thus, the standard form of the equation for a hyperbola with focal points on the $x$ axis is:
  • If the focal points are on the $y$-axis, the variables simply change places:
  • The ellipse can be defined as all points that have a constant sum of distances to two focal points, and the hyperbola is defined as all points that have constant difference of distances to two focal points.

• Parts of a Hyperbola

  • Therefore the focal points are located at $(h+2\sqrt{m},k+2\sqrt{m})$ and $(h-2\sqrt{m},k-2\sqrt{m})$.

• Types of Conic Sections

  • A vertex, which is the point at which the curve turns around
  • Its intersection with the cone is therefore a set of points equidistant from a common point (the central axis of the cone), which meets the definition of a circle.
  • A radius, which the distance from any point on the circle to the center point
  • Two focal points—for any point on the ellipse, the sum of the distances to both focal points is a constant
  • Two focal points, around which each of the two branches bend

• Parabolas As Conic Sections

  • The vertex is therefore also a point on the cone, and the distance between that point and the cone's central axis is the radius of a circle.
  • The focal length is the leg of the right triangle that exists along the axis of symmetry, and the focal point is the vertex of the right triangle.
  • Using the definition of sine as opposite over hypotenuse, we can find a formula for the focal length "$f$" in terms of the radius and the angle:
  • The vertex will be at the point:
  • A right triangle is formed from the focal point of the parabola.

• Applications of Hyperbolas

  • A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals between the point and the given points.
  • Trilateration is the a method of pinpointing an exact location, using its distances to a given points.
  • The can also be characterized as the difference in arrival times of synchronized signals between the desired point and known points.
  • One way of defining a hyperbola is as precisely this: the curve of points such that the absolute value of the difference between the distances to two focal points remains constant.
  • Orbits which are circular or elliptical are bound orbits, which is to say the object never escapes its closed path around one of the focal points.

• Parts of an Ellipse

  • The foci are two points inside the ellipse that characterize its shape and curvature.
  • For a horizontal ellipse, the foci have coordinates $(h \pm c,k)$, where the focal length $c$ is given by
  • This diagram of a horizontal ellipse shows the ellipse itself in red, the center $C$ at the origin, the focal points at $\left(+f,0\right)$ and $\left(-f,0\right)$, the major axis vertices at $\left(+a,0\right)$ and $\left(-a,0\right)$, the minor axis vertices at $\left(0,+b\right)$ and $\left(0,-b\right)$.
  • It also shows how the sum of the distances from any point on the ellipse to the two foci is a constant, and how the eccentricity is determined by relating one of the foci to a line $D$ called the directrix.

• Standard Equations of Hyperbolas

  • In this special case, the rectangle joining the four points on the asymptotes directly above and below the vertices is a square, since the lengths of its sides 2a = 2b.
  • The two focal points are labeled F1 and F2, and the thin black line joining them is the transverse axis.
  • 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).

• Applications of the Parabola

  • One well-known example is the parabolic reflector—a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point.
  • Conversely, a parabolic reflector can collimate light from a point source at the focus into a parallel beam.

• Point-Slope Equations

  • The point-slope form is ideal if you are given the slope and only one point, or if you are given two points and do not know what the $y$-intercept is.
  • Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
  • Then plug this point into the point-slope equation and solve for $y$ to get:
  • Example: Write the equation of a line in point-slope form, given point $(-3,6)$ and point $(1,2)$, and convert to slope-intercept form
  • Plug this point and the calculated slope into the point-slope equation to get:

• The Distance Formula and Midpoints of Segments

  • The distance and the midpoint formulas give us the tools to find important information about two points.
  • The distance can be from two points on a line or from two points on a line segment.  
  • The distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by the formula:
  • In geometry, the midpoint is the middle point of a line segment, or the middle point of two points on a line, and thus is equidistant from both end-points.
  • If you have two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the midpoint of the segment connecting the two points can be found with the formula: