Parallel lines are lines that do not intersect no matter how long they extend. In analytical geometry, two lines are said to be parallel if and only if their slopes are the same.

Applications:

**Problem 1:**

Find the equation of a line through (1,2) and parallel to 2x-y+1=0.

Solution:

2x-y+1=0 when re arrange to slope-intercept form is y=2x+1 with slope 2.

To get the required equation,

y-y_{1}=m(x-x_{1}), where m=2(same slope) and point (1,2)

By substitution,

y-2=2(x-1)

y-2=2x-2

**2x-y=0** is the required equation.

**Problem 2:**

What is the shortest distance between lines 2x-y+1=0 and 2x-y=0?

We already know that these are parallel lines in problem 1.

**Solution 1:**

The shortest distance between these lines is the distance of perpendicular segment drawn from one line to the other line.

Select any point on any line, let’s choose point (1,2) on 2x-y=0 and use the distance of a point to a line formula. We are now looking for the perpendicular distance between 2x-y+1=0 from (1,2)

the sign of denominator will follow the sign of numerator

*d***=****√****3/3**

**Solution 2:**

The shortest distance between parallel lines ax+by+c_{1} and ax+by+c_{2} can be calculated using the formula below. Given 2x-y+1 and 2x-y=0, c_{2}=0, c_{1}=1

the sign of the denominator will follow the sign of numerator.

*d***=****√****3/3**

Practice problems

1. Find the equation of the line in standard form through (2,-1) parallel to 3x-2y=4.

2. What the shortest distance between the lines 3x-5y-4=0 and 6x-10y+4=0?

3. Find the equation of the line through (-1,4) and parallel to 5x-2y-3=0

4. Find the shortest distance between the lines x-2y-5=0 and 3x-6y-2=0

5. The line 2x-ky=4 and x-y=5 are parallel lines. What is the value of k?

Answer key here