Introduction to Perpendicular Lines

Perpendicular lines are a fundamental concept in geometry and algebra, and understanding how to find the equation of a line that is perpendicular to a given line and passes through a specific point is crucial. This article will guide you through the process, providing clear examples and explanations.

Understanding the Concept of Perpendicular Lines

Perpendicular lines are lines that intersect at a 90-degree angle. The slope of one line is the negative reciprocal of the slope of the other line. This means if the slope of one line is ( m ), the slope of a line perpendicular to it will be ( -frac{1}{m} ).

Step-by-Step Guide

To find the equation of a line that is perpendicular to a given line and passes through a specific point, follow these steps:

Step 1: Determine the slope of the given line

Let's consider a given line with the equation ( y -2x^2 ). The slope of this line is -2 because in the general form of a linear equation ( y mx b ), the coefficient ( m ) represents the slope. So, for ( y -2x^2 ), the slope ( m_1 -2 ).

Step 2: Calculate the slope of the perpendicular line

The slope of the perpendicular line, ( m_2 ), is the negative reciprocal of the slope of the given line. Therefore, ( m_2 -frac{1}{-2} frac{1}{2} ).

Step 3: Write the point-slope form of the perpendicular line

The point-slope form of a line's equation is given by ( y - y_1 m(x - x_1) ), where ( (x_1, y_1) ) is a point on the line and ( m ) is the slope. Given that the line passes through the point ( (0, 1) ), and the slope ( m_2 frac{1}{2} ), the equation becomes:

[ y - 1 frac{1}{2}(x - 0) ]

Simplifying this, we get:

[ y frac{1}{2}x 1 ]

Alternatively, you can rewrite this equation in standard form:

[ x - 2y 2 0 ]

Step 4: Derive the equation algebraically

Given the line ( y -2x^2 ), we want a line perpendicular to it that passes through the point ( (0, 1) ). The slope of this perpendicular line is ( frac{1}{2} ). Using the point-slope form:

[ y - 1 frac{1}{2}(x - 0) ]

Simplifying, we get:

[ y frac{1}{2}x 1 ]

And in standard form:

[ x - 2y 2 0 ]

Conclusion

Finding the equation of a line that is perpendicular to another given line and passes through a specific point involves determining the slope of the given line, calculating the slope of the perpendicular line, and using the point-slope form of the line's equation. By following these steps, one can easily derive the equation of the desired line.

For further practice, try deriving the equation of a line that is perpendicular to ( y 5x ) and passes through the point ( (2, 3) ). Good luck!