Step By Step Calculus » 4.2 - Lines

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Synopsis
A straight line in the plane is the set of points satisfying an equation of the form
\displaystyle \{(x, y) \, \mid \, Ax + By + C =0 \},
\displaystyle \{(x, y) \, \mid \, Ax + By + C =0 \},
where A, \, B, \, CA, \, B, \, C are given constants.

In slope-intercept form, the equation of a straight line is y=mx+by=mx+b where mm is the slope and bb is the yy-intercept. This is the default form you should put your answers in unless otherwise stated. The only line whose equation cannot be written in this form is a vertical line: x = k x = k .
In point-slope form, the equation of a straight line is y-y_1=m(x-x_1)y-y_1=m(x-x_1) where mm is the slope and (x_1,y_1)(x_1,y_1) is a given point on the line. Given two points ( x_1,y_1 )( x_1,y_1 ) and (x_2, y_2 ) (x_2, y_2 ) on a line, the slope can be expressed as
\displaystyle m= \frac { y_{2} - y_{1} }{ x_{2} - x_{1} } = \frac { \textrm{rise}}{ \textrm{run} }.
\displaystyle m= \frac { y_{2} - y_{1} }{ x_{2} - x_{1} } = \frac { \textrm{rise}}{ \textrm{run} }.
Two lines with respective slopes m_1m_1 and m_2m_2 are parallel if and only if m_1=m_2m_1=m_2. They are perpendicular if and only if m_1m_2=-1.m_1m_2=-1. In addition vertical and horizontal lines are obviously perpendicular, even though the slope of the vertical line is undefined.
The projection of a point PP on a Line LL is a point QQ on LL such that PQPQ is perpendicular to LL.
The line passing through two given points (x_1,y_1)(x_1,y_1) and (x_2,y_2)(x_2,y_2) on a parabola is called a secant line. This secant line transforms to a tangent line at (x_1, y_1)(x_1, y_1) as the other point (x_2,y_2)(x_2,y_2) approaches (x_1,y_1)(x_1,y_1). Thus, the slope of the tangent line is \displaystyle m=\lim_{x_2\to x_1} \frac{y_2-y_1}{x_2-x_1} = (2ax_1+b)\displaystyle m=\lim_{x_2\to x_1} \frac{y_2-y_1}{x_2-x_1} = (2ax_1+b) and the equation of tangent line at (x_1,y_1)(x_1,y_1) is y-y_1=m(x-x_1)\Longleftrightarrow y=(2ax_1+b)x-ax_1^2+c.y-y_1=m(x-x_1)\Longleftrightarrow y=(2ax_1+b)x-ax_1^2+c. One major accomplishment of the calculus is to determine a formula for the slope of the tangent line by developing a theory of limits, not only for parabolas, but for general curves in any dimension.
If the graph of the straight line y=mx+by=mx+b lies above the xx-axis between x=ax=a and x=\overline{x}x=\overline{x}, then the area bounded by the graph, the xx-axis and the vertical lines x = ax = a, x = \overline{x}x = \overline{x} is a trapezoid. Its area AA can be computed using the area formula for a trapezoid :
\displaystyle A=\frac{1}{2}m\overline{x}^2+b\overline{x}-\frac{1}{2}ma^2-ba.
\displaystyle A=\frac{1}{2}m\overline{x}^2+b\overline{x}-\frac{1}{2}ma^2-ba.