What is the shortest distance from the point (x_1, y_1) to the straight line, Ax + By + C = 0?

Question:

What is the shortest distance from the point (x{eq}_1 {/eq}, y{eq}_1 {/eq}) to the straight line, Ax + By + C = 0?

Coordinate Geometry:

The following problem deals with the concept of Coordinate Geometry, which was introduced by Rene Descartes. The perpendicular distance is always the shortest distance between any two objects.

Answer and Explanation:

The perpendicular distance of a point (a,b) from the line kx+ly+m=0 is {eq}\displaystyle \boxed{\mathbf{\frac{|ak+bl+m|}{\sqrt{k^2+l^2}}}} {/eq}

Plugging in the variables, we get

{eq}\displaystyle \begin{align} \text{Shortest distance} &= \frac{Ax_1+By_1+C}{\sqrt{A^2+B^2}} \end{align} {/eq}

Therefore, the shortest distance from the point (x_1, y_1) to the straight line, Ax + By + C = 0 is {eq}\displaystyle \boxed{\mathbf{ \frac{Ax_1+By_1+C}{\sqrt{A^2+B^2}}}} {/eq}.


Learn more about this topic:

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How to Find the Distance Between a Point & a Line

from FTCE Mathematics 6-12 (026): Practice & Study Guide

Chapter 30 / Lesson 5
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