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Chapter 0: Problem 88

Distance Write the distance \(d\) between the point \((3,1)\) and the line \(y=m x+4\) in terms of \(m .\) Use a graphing utility to graph the equation. When is the distance 0\(?\) Explain the result geometrically.

Short Answer

Expert verified
The distance \(d\) between the point (3,1) and the line \(y=mx+4\) in terms of \(m\) is \[d = \frac{|-3m - 3|}{\sqrt{m^2+1}}\] From the graph, the distance is zero when the point (3,1) lies on the line \(y=mx+4\), which geometrically implies that for those value of slopes, the line will pass through the point.

Step by step solution

01

Define the Distance from a Point to a Line

The distance \( d \) from a point \( (x_1, y_1) \) to a line \( ax + by + c = 0 \) is given by the formula \[d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2+b^2}}\] We will use this formula to find the distance from the point (3,1) to the line \(y=mx+4\).
02

Convert the Line Equation

First, convert the given line equation \(y = mx + 4\) to standard form \(ax + by + c = 0 \). This gives \[-mx + y - 4 = 0\] Now we can equate it in the formula, where \(a=-m, b=1, c=-4, x_1=3 \) and \( y_1 = 1 \).
03

Substitute the Point and Line into the Distance Formula

Substituting these values into the distance formula gives: \[d = \frac{|-m*3 + 1 - 4|}{\sqrt{(-m)^2+1^2}}\] This simplifies to \[d = \frac{|-3m - 3|}{\sqrt{m^2+1}}\]
04

Graph the Equation and Identify where Distance is Zero

Graph the expression for \(d\) as a function of \(m\). Where the graph intersects the \(m\)-axis (i.e. where \(d(m) = 0\)), that's the slope at which distance between the point and the line is zero. This geometrically implies that the point lies on the line.

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