求过一点的垂线的方程

It is a common situation in coordinate geometry to have the equation of a line and the coordinates of a point where it is desired to find the equation of the line perpendicular to the given line that goes through the given point. With the equation of the perpendicular line, we can find the distance from the given point to the given line as well.

The easiest way to illustrate the steps of solving such a problem is by using an example. So, let's say we have a line $y = 4x+6$ and a point $(6,8)$. We want to find the distance from the point to this line.

We start by finding the line perpendicular to $y = 4x+6$ that passes through $(6,8).$

So if $m$ is the slope of our perpendicular line, we have $4m = -1 \implies m = -\frac14$. Then, writing our line in point-slope form gives $$y - 8 = -\frac14(x-6) \implies 32 - 4y = x - 6 \implies y = -\frac{x}{4}+\frac{19}{2}.$$ The intersection point of the two perpendicular lines then is given by $$-\frac{x}{4} + \frac{19}{2} = 4x + 6 \implies -x + 38 =16x + 24 \implies x = \frac{14}{17}.$$ So $y = 4*\frac{14}{17} + 6 = \frac{158}{17}.$ Now we just use the distance formula to find the distance between points $(6,8)$ and $(\frac{14}{17},\frac{158}{17})$. Since I (unintentionally) picked bad numbers for the example, this is left for the interested reader as an exercise.

In fact, there is a formula that exists to find the distance between a point and a line to avoid computations like those above. For a line given by the equation $ax+by+c = 0,$ its distancce from a point $(p,q)$ is given by: $$\frac{\left|ap+bq+c\right|}{\sqrt{a^2+b^2}}$$ The formula can be proved by manually finding the equation of the perpendicular line and using the distance formula in terms of the variables.

(Video) Equation of a Line Perpendicular to Another Line

by Math Meeting

In this video by Math Meeting it is explained how to find the equation of a line perpendicular to another line.

(Video) Perpendicular Lines

by The Math Sorcerer

In the above few videos from The Math Sorcerer it is shown how to find the equation of a line perpendicular to another.