parallel lines

Examples of parallel lines in the following topics:

• Parallel and Perpendicular Lines

  • Two lines in a plane that do not intersect or touch at a point are called parallel lines.
  • For example, given two lines: $f(x)=m_{1}x+b_{1}$and $g(x)=m_{2}x+b_{2}$, writing $f(x)$ $\parallel$ $g(x)$ states that the two lines are parallel to each other.  
  • In 2D, two lines are parallel if they have the same slope.
  • Given two parallel lines $f(x)$ and $g(x)$, the following is true:
  • Write equations for lines that are parallel and lines that are perpendicular

• Inconsistent and Dependent Systems in Two Variables

  • Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line.
  • Thus, the two lines are dependent.
  • When graphed, the two equations produce identical lines, as demonstrated below.
  • Recall that the graphical representation of an inconsistent system consists of parallel lines that have the same slope but different $y$-intercepts.
  • The graphs of these equations on the $xy$-plane are a pair of parallel lines.

• The Vertical Line Test

  • To use the vertical line test, take a ruler or other straight edge and draw a line parallel to the $y$-axis for any chosen value of $x$.
  • If, alternatively, a vertical line intersects the graph no more than once, no matter where the vertical line is placed, then the graph is the graph of a function.
  • For example, a curve which is any straight line other than a vertical line will be the graph of a function.
  • The vertical line test demonstrates that a circle is not a function.
  • Any vertical line in the bottom graph passes through only once and hence passes the vertical line test, and thus represents a function.

• Standard Equations of Hyperbolas

  • At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center.
  • The asymptotes of the hyperbola (red curves) are shown as blue dashed lines and intersect at the center of the hyperbola, C.
  • The two focal points are labeled F1 and F2, and the thin black line joining them is the transverse axis.
  • The perpendicular thin black line through the center is the conjugate axis.
  • The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2.

• Asymptotes

  • In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity.
  • Horizontal asymptotes are parallel to the $x$-axis.
  • Vertical asymptotes are vertical lines near which the function grows without bound.
  • They are parallel to the $y$-axis.
  • These are diagonal lines so that the difference between the curve and the line approaches $0$ as $x$ tends to $+ \infty$ or $- \infty$.

• Slope-Intercept Equations

  • One of the most common representations for a line is with the slope-intercept form.
  • Note that if $m$ is $0$, then $y=b$ represents a horizontal line.
  • However, a vertical line is defined by the equation $x=c$ for some constant $c$.
  • This assists in finding solutions to various problems, such as graphing, comparing two lines to determine if they are parallel or perpendicular and solving a system of equations.
  • Thus we arrive at the point $(2,-5)$ on the line.

• Parabolas As Conic Sections

  • In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.
  • The axis of symmetry is a line that is at the same angle as the cone and divides the parabola in half.
  • The directrix is a straight line on the opposite side of the parabolic curve from the focus.
  • Conversely, light that originates from a point source at the focus is reflected, or collimated, into a parallel beam.
  • The light leaves the parabola parallel to the axis of symmetry.

• Types of Conic Sections

  • This creates a straight line intersection out of the cone's diagonal.
  • A circle is formed when the plane is parallel to the base of the cone.
  • The general form of the equation of an ellipse with major axis parallel to the x-axis is:
  • The general equation for a hyperbola with vertices on a horizontal line is:
  • The other degenerate case for a hyperbola is to become its two straight-line asymptotes.

• Inconsistent and Dependent Systems in Three Variables

  • The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
  • An example of three different equations that intersect on a line.
  • (b) Two of the planes are parallel and intersect with the third plane, but not with each other.
  • (c) All three planes are parallel, so there is no point of intersection.
  • Two equations represent the same plane, and these intersect the third plane on a line.

• Applications of Hyperbolas

  • A hyperbola is an open curve with two branches and a cut through both halves of a double cone, which is not necessarily parallel to the cone's axis.
  • The plane may or may not be parallel to the axis of the cone.
  • If we mark where the end of the shadow falls over the course of the day, the line traced out by the shadow forms a hyperbola on the ground (this path is called the declination line).
  • In the figure, the blue line shows the hyperbolic Kepler orbit.
  • The plane may or may not be parallel to the axis of the cone.