Parallel Line Through a Point Calculator

What is a Parallel Line Through a Point?

Imagine you have a straight line on a piece of paper. Now, think about drawing another line that never touches the first one, no matter how far you extend both lines. This new line is called a parallel line. When we say "through a point," we mean that this parallel line passes through a specific dot on your paper. It's like drawing a railroad track parallel to an existing one, but making sure it goes through a particular spot you've marked!

How to Calculate a Parallel Line Through a Point

To find a parallel line through a point, we start with the equation of the original line and the coordinates of our special point. Then, we use these to create a new equation for our parallel line. It's like following a recipe to create a twin line that goes through our chosen spot!

Formula

We use the general form of a line equation: \(Ax + By + C = 0\)

For our parallel line, we keep A and B the same, but change C:

\[ Ax + By + D = 0 \]

Where:

• \(A\) and \(B\) are the same numbers from the original line equation
• \(D\) is a new number we calculate to make the line go through our point
• \(x\) and \(y\) are the coordinates of any point on the new line

Calculation Steps

1. Write down the equation of the original line: \(Ax + By + C = 0\)
2. Note the coordinates of your point: \((x_0, y_0)\)
3. Calculate D using this formula: \(D = -(Ax_0 + By_0)\)
4. Write the new equation: \(Ax + By + D = 0\)
5. This is your parallel line through the point!

Example and Visual Representation

Let's find a line parallel to \(2x - y + 1 = 0\) that passes through the point (3, 4).

Following our steps:

1. Original line: \(2x - y + 1 = 0\)
2. Our point: (3, 4)
3. Calculate D: \(D = -(2(3) - 4) = -2\)
4. New equation: \(2x - y - 2 = 0\)

Let's see this on a graph:

In this picture, the red line is our original line, and the blue line is our new parallel line. The green dot shows our point (3, 4). See how the blue line passes through the green dot while staying parallel to the red line? It's like we've drawn a new train track that's always the same distance from the original one, but it goes right through our special station at (3, 4)!