Understanding the Perpendicular Line Equation Through a Specific Point

The equation of a straight line passing through a given point and perpendicular to a given line is a fundamental problem in coordinate geometry. In this article, we will derive the equation of a straight line that is perpendicular to the line 3x - 5y - 1 0 and passes through the point (0, 2).

Deriving the Perpendicular Line

The first step in solving this problem is to determine the slope of the given line, 3x - 5y - 1 0.

Step 1: Converting to Slope-Intercept Form

Let's start by expressing 3x - 5y - 1 0 in the slope-intercept form (y mx b), where m is the slope of the line:

3x - 5y - 1 0

Move the terms involving y to one side and the rest to the other side:

3x - 1 5y

Divide both sides by 5 to solve for y:

y -3/5x 1/5

The slope of the given line is -3/5.

Step 2: Determining the Slope of the Perpendicular Line

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope of the line perpendicular to -3/5 is:

5/3

Step 3: Applying the Point-Slope Formula

The point-slope formula for a line through a point (x1, y1) with slope m is:

y - y1 m(x - x1)

Using the point (0, 2) and the slope 5/3, we can derive the equation of the perpendicular line:

   y - 2  (5/3)(x - 0)   y - 2  5/3x   y  5/3x   2

Simplifying, the equation of the line is:

y 5/3x 2

Alternative Derivation Methods

There are other methods to find the equation of the line. Here are two additional approaches:

Method 1: General Form Approach

The general form of a line that is perpendicular to 3x - 5y - 1 0 can be written as:

5x - 3y k 0

To find the value of k, we use the point (0, 2):

5(0) - 3(2)   k  0-6   k  0k  6

Therefore, the equation of the perpendicular line is:

5x - 3y 6 0

Method 2: Simplified Approach

Starting with the given line:

3x - 5y - 1 0

Rewriting it in slope-intercept form:

y -3/5x 1/5

The slope of the original line is -3/5. The slope of the perpendicular line is 5/3. Using the point (0, 2) and the slope 5/3:

y - 2  (5/3)(x - 0)y - 2  5/3xy  5/3x   2

Simplifying, the equation of the line is:

y 5/3x 2

Conclusion

In this article, we have demonstrated three methods to find the equation of a line perpendicular to 3x - 5y - 1 0 and passing through the point (0, 2). These methods include using the point-slope formula, the general form of the equation, and a simplified approach.

The final equation of the line is:

y 5/3x 2