Introduction to Finding Parallel Lines at a Given Distance

The concept of finding a line parallel to a given line at a specified distance is a fundamental aspect of coordinate geometry. In this article, we will explore the method to determine such lines, specifically focusing on the parallel lines to the equation 5x - 12y 0 that are 5 units apart from it.

Understanding the Equation of the Original Line

Step 1: Identifying the Slope

The equation of the original line can be expressed as 5x - 12y 0. To find its slope, we rewrite it in the slope-intercept form y mx b:

12y 5x y 512x

Hence, the slope m is -512.

Constructing Parallel Lines

Parallel lines have the same slope. Thus, any line parallel to the original line will have the equation in the form:

5x - 12y c

where c is a constant that we need to determine.

Calculating Distance Between Lines

Step 2: Calculating the Distance to the Parallel Line

The distance d from a point (x_0, y_0) to a line given by the equation Ax By C 0 is calculated using the formula:

d |Ax_0 By_0 C|sqrt{A^2 B^2}

For the line 5x - 12y 0, we have A 5, B -12, and C 0. The distance from the original line to the new line can be expressed as:

d |c|sqrt{5^2 12^2} |c|13

To find the lines that are 5 units away from the original line, we set d equal to 5:

|c|13 5 implied c plusmn;65

Final Equations of Parallel Lines

Therefore, the equations of the lines parallel to 5x - 12y 0 and 5 units away are:

5x - 12y 65 and 5x - 12y -65

Another Geometric Interpretation

Consider a line y kx S and a parallel line and a perpendicular connector of them having length D. If the angle of inclination (alsa) is given by tan(alfa) k, the distance S between the lines can be determined using:

C cos(alfa) 1 - (k)^2/1 (k)^2

For the line with tan(alfa) -5/12, we have:

C cos(alfa) 1 - (5/12)^2 0.726025

The equation of the desired line is thus:

y -5/12x - 5/C or y -5/12x - 7.691

Since the shift can be either upward or downward, the final equations for the parallel lines are:

5x - 12y -65 and 5x - 12y 65

Conclusion

In summary, to find parallel lines at a specified distance from a given line, we established the slope, wrote the equation in the required form, calculated the distance, and determined the values of the constant c. This method not only applies to the example provided but can be generalized to other cases of coordinate geometry.