perpendicular lines

Examples of perpendicular lines in the following topics:

• Parallel and Perpendicular Lines

  • Two lines are perpendicular to each other if they form congruent adjacent angles.
  • This means that if the slope of one line is $m$, then the slope of its perpendicular line is $\frac{-1}{m}$.
  • Therefore, the equation of the line perpendicular to the given line has a slope of $-4$ and a $y$-intercept of $12$.  
  • The line $f(x)=3x-2$ in red is perpendicular to line $g(x)=\frac{-1}{3}x+1$ in blue. 
  • Write equations for lines that are parallel and lines that are perpendicular

• 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 distance b (not shown in below) is the length of the perpendicular segment from either vertex to the asymptotes.
  • 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.

• 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.
  • 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.  

• 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.

• What Are Conic Sections?

  • If the plane is parallel to the generating line, the conic section is a parabola.
  • If the plane is perpendicular to the axis of revolution, the conic section is a circle.
  • A directrix is a line used to construct and define a conic section.
  • These distances are displayed as orange lines for each conic section in the following diagram.
  • A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix.

• Eccentricity

  • The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest 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.
  • Eccentricity is the ratio between the distance from any point on the conic section to its focus, and the perpendicular distance from that point to the nearest directrix.

• Introduction to Hyperbolas

  • The intersection of a right circular double cone with a plane perpendicular to the base of the cone
  • The set of all points such that the ratio of the distance to a single focal point divided by the distance to a line (the directrix) is greater than one

• The Distance Between Two Lines

• The Existence of Inverse Functions and the Horizontal Line Test

  • Recognize whether a function has an inverse by using the horizontal line test

• 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.