Find the perpendicular line of y = -2x + 4 that passes through the point (0,0).
Understand the Problem
The question is asking to find the equation of a line that is perpendicular to the given line y = -2x + 4 and also passes through the point (0,0). To solve this, we need to identify the slope of the given line, find the negative reciprocal of that slope for the perpendicular line, and then use the point-slope form of the equation of a line to derive the equation for the new line.
Answer
The equation of the line is $y = \frac{1}{2}x$.
Answer for screen readers
The equation of the line that is perpendicular to the given line and passes through the point (0,0) is
$$ y = \frac{1}{2}x $$
Steps to Solve
- Identify the slope of the given line
The equation of the given line is in the slope-intercept form ( y = mx + b ), where ( m ) is the slope. For the line ( y = -2x + 4 ), the slope ( m ) is ( -2 ).
- Find the slope of the perpendicular line
To find the slope of the line that is perpendicular to the given line, we need to take the negative reciprocal of the original slope. The negative reciprocal of ( -2 ) is
$$ \frac{1}{2} $$
- Use the point-slope form to find the equation of the new line
We use the point-slope form of the equation of a line, which is
$$ y - y_1 = m(x - x_1) $$
where ( (x_1, y_1) ) is the point the line passes through, and ( m ) is the slope. Substituting the point ( (0,0) ) and the slope ( \frac{1}{2} ):
$$ y - 0 = \frac{1}{2}(x - 0) $$
- Simplify the equation
Now, we can simplify the equation:
$$ y = \frac{1}{2}x $$
The equation of the line that is perpendicular to the given line and passes through the point (0,0) is
$$ y = \frac{1}{2}x $$
More Information
The line $y = \frac{1}{2}x$ has a slope of $\frac{1}{2}$, which means that for every 2 units it moves horizontally to the right, it moves 1 unit up vertically. This is a basic concept of perpendicular lines in coordinate geometry; their slopes multiply to -1.
Tips
- Forgetting to take the negative reciprocal of the slope when finding the perpendicular slope.
- Misapplying the point-slope form of the line equation.
AI-generated content may contain errors. Please verify critical information
