Question

line l contains the points (4,2) and (12,4). find the distance between the line and the point (6,11). leave your answer in radical form. distance =

Explanation:

Step1: Find the equation of line \(l\)

Use the two - point form \(y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\). Given \((x_1,y_1)=(4,2)\) and \((x_2,y_2)=(12,4)\), the slope \(m=\frac{4 - 2}{12 - 4}=\frac{2}{8}=\frac{1}{4}\).
The equation of the line is \(y - 2=\frac{1}{4}(x - 4)\), which simplifies to \(x-4y + 4 = 0\).

Step2: Use the distance formula

The distance \(d\) from a point \((x_0,y_0)\) to a line \(Ax+By + C = 0\) is given by \(d=\frac{\vert Ax_0+By_0 + C\vert}{\sqrt{A^{2}+B^{2}}}\).
Here, \(A = 1\), \(B=-4\), \(C = 4\), \(x_0=6\), \(y_0 = 11\).
Substitute into the formula: \(d=\frac{\vert1\times6+( - 4)\times11 + 4\vert}{\sqrt{1^{2}+(-4)^{2}}}=\frac{\vert6-44 + 4\vert}{\sqrt{1 + 16}}=\frac{\vert-34\vert}{\sqrt{17}}\).
Since \(\vert-34\vert=34\), then \(d=\frac{34}{\sqrt{17}}\).
Rationalize the denominator: \(d = 2\sqrt{17}\).

Answer:

\(2\sqrt{17}\)