Distance from a Line to the Origin: A Comprehensive Guide

Understanding how to calculate the distance between a line and the origin in the coordinate plane is a fundamental skill in mathematics, particularly in geometry and algebra. This problem often arises in various applications, including computer graphics, physics, and engineering. In this article, we will walk through a detailed solution to the problem: finding the distance between the line 3x 4y 5 and the origin (0,0).

Problem Statement

Given the line 3x 4y 5, we want to find the distance between this line and the origin (0,0).

Step-by-Step Solution

Step 1: Find the Equation of the Perpendicular Line Through the Origin

The distance between a line and a point can be found by using the perpendicular distance formula. To start, we need to find the equation of the line that passes through the origin (0,0) and is perpendicular to the given line 3x 4y 5.

First, convert the given line into the slope-intercept form: 4y -3x 5 or y -3/4x 5/4. The slope of this line is -3/4. The slope of the perpendicular line is the negative reciprocal, which is 4/3.

Step 2: Determine the Equation of the Perpendicular Line Through the Origin

The equation of the perpendicular line through the origin, using the slope-intercept form y mx b, is simply y 4/3x because the line passes through the origin and the y-intercept (b) is 0.

Step 3: Find the Intersection of the Two Lines

To find the distance, we need to determine where the two lines intersect. We set up a system of equations and solve for x and y.

The given line is 3x 4y 5 and the perpendicular line is y 4/3x.

Substitute y from the perpendicular line into the given line equation:

3x 4(4/3x) 5

3x 16/3x 5

Multiply through by 3 to clear the fraction:

9x 16x 15

25x 15

x 15/25 3/5

Substitute x 3/5 back into the equation y 4/3x:

y 4/3(3/5) 4/5

Thus, the two lines intersect at the point (3/5, 4/5).

Step 4: Calculate the Distance Using the Distance Formula

The distance between the point (3/5, 4/5) and the origin (0,0) can be found using the distance formula:

d √[(x2 - x1)^2 (y2 - y1)^2]

Substitute (0,0) for (x1, y1) and (3/5, 4/5) for (x2, y2):

d √[(3/5 - 0)^2 (4/5 - 0)^2]

d √[(3/5)^2 (4/5)^2]

d √[9/25 16/25]

d √[25/25] √1 1

Thus, the distance from the line 3x 4y 5 to the origin is 1 unit.

Verifying the Result

Another way to verify this result is by using the squared distance formula:

d^2 (3x 4y - 5)^2 / (3^2 4^2)

Substitute (0,0) into the formula:

d^2 (3(0) 4(0) - 5)^2 / (3^2 4^2) (0 - 5)^2 / (9 16) 25 / 25 1

This confirms that the squared distance is 1, so the distance is 1 unit.

Another method is to recognize that the point (3/5, 4/5) lies on the unit circle, as (3/5)^2 (4/5)^2 1. The coefficients of the line 3x/5 4y/5 1 indicate that it is tangent to the unit circle at this point. Hence, the distance is 1, the radius of the unit circle.

Conclusion

The distance between the line 3x 4y 5 and the origin is a classic problem in geometry, and solving it involves understanding the properties of lines and circles. By following the steps laid out above, we can find that the distance is 1 unit. This problem reinforces the importance of knowing how to use the distance formula and the equations of lines and circles.

Key takeaways:

Distance from a line to the origin can be calculated using the perpendicular distance formula. Perpendicular lines and their properties are crucial for solving such problems. Distance formula and the properties of circles are essential tools in geometric problem-solving.