Q. 6TB - Calculus

Question

The distance between a point and a line in the plane: Describe a method for computing the distance between the point \(\left ( x_{0},y_{0} \right )\) and the line y = mx + b.

Step-by-Step Solution

Answer

The distance between the point \(\left ( x_{0},y_{0} \right )\) and the line y = mx + b in the plane is \(d=\frac{\left|mx_{0}-y_{0}+b \right|}{\sqrt{x_{0}^{2}+y_{0}^{2}}}\).

Step 1. Given Information

The given points are \(\left ( x_{0},y_{0} \right )\) and the line is y = mx + b. 

We have to describe a method for computing the distance between the point and the line.

Step 2. Describing a method for computing the distance between the point and the line

To find the distance between the point and the line we will use the formula \(d=\frac{\left|Ax_{1}+By_{1}+C \right|}{\sqrt{A^{2}+B^{2}}}\).

To apply this formula, we will have to express the line in standard form

\(y=mx+b\)

\(mx-y+b=0\)

Thus, \(A=m, B=-1, C=b\) and the points are \(\left ( x_{0},y_{0} \right )\) .

Therefore,

\(d=\frac{\left|m\cdot x_{0}+(-1)\cdot y_{0}+b \right|}{\sqrt{x_{0}^{2}+y_{0}^{2}}}\)

\(d=\frac{\left|mx_{0}-y_{0}+b \right|}{\sqrt{x_{0}^{2}+y_{0}^{2}}}\).