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Chapter 0: Problem 87

Distance Show that the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+C=0\) is Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\)

Short Answer

Expert verified
The distance between a given point \((x_{1}, y_{1})\) and the line \(Ax + By + C = 0\) is indeed given by \(\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\) as proven by the substitution method into the equation of the line.

Step by step solution

01

Analyzing the equation of the line

The general equation of a line in the 2-D coordinate plane is given as \(Ax + By + C = 0\), where A, B and C are real numbers. We can rewrite this equation in the slope-intercept form \(y = mx + c\), where m (slope of the line) = \(-\frac{A}{B}\) and c (y-intercept) = \(-\frac{C}{B}\).
02

Determining the formula for the distance between a point and a line

The distance \(d\) between a point \(P(x_{1},y_{1})\) and a line given by the equation \(Ax + By + C = 0\) can be determined by the formula \(d = \frac{\left|A x_{1} + B y_{1} + C\right|}{\sqrt{A^{2} + B^{2}}}\). The numerator is the absolute value of the substitution of the coordinates of the point into the equation of the line, and the denominator is the square root of the sum of the squares of the coefficients of x and y.
03

Verifying the formula

Now to prove our formula is correct, take any point \(P(x_{1},y_{1})\) and line \(Ax + By + C = 0\) and calculate the distance using the formula. If the formula holds, then we have successfully proved it.

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Most popular questions from this chapter

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.

Even and Odd Functions and Zeros of Functions In Exercises \(79-82\) , determine whether the function is even, odd, or neither. Then find the zeros of the function. Use a graphing utility to verify your result. $$ f(x)=x \cos x $$

Finding the Domain and Range of a Function In Exercises \(11-22,\) find the domain and range of the function. $$ f(x)=|x-3| $$

Distance In Exercises \(83-86\) , find the distance between the point and line, or between the lines, using the formula for the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+\) \(C=0 .\) $$ =\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}} $$ Line: \(x+y=1\) Line: \(x+y=5\)

Finding the Domain and Range of a Function In Exercises \(11-22,\) find the domain and range of the function. $$ h(x)=-\sqrt{x+3} $$
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