Find the perpendicular line of 4x+y=2 that passes through the point (1,1)
Understand the Problem
The question is asking for the equation of a line that is perpendicular to the given line (4x + y = 2) and passes through the point (1, 1). To solve this, we'll first determine the slope of the given line, then find the negative reciprocal of that slope to get the slope of the perpendicular line. Finally, we'll use the point-slope form of a line equation with the new slope and the point (1, 1).
Answer
The equation of the perpendicular line is $y = \frac{1}{4}x + \frac{3}{4}$.
Answer for screen readers
The equation of the line is given by:
$$ y = \frac{1}{4}x + \frac{3}{4} $$
Steps to Solve
- Find the slope of the given line
First, we need to rewrite the given equation of the line, $4x + y = 2$, in slope-intercept form $y = mx + b$, where $m$ is the slope.
Rearranging the equation gives:
$$ y = -4x + 2 $$
So, the slope of the given line is $m = -4$.
- Determine the slope of the perpendicular line
The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope.
The negative reciprocal of $-4$ is:
$$ m_{\text{perpendicular}} = \frac{1}{4} $$
- Use point-slope form to find the equation of the perpendicular line
Now, we can use the point-slope form of a line equation, which is:
$$ y - y_1 = m(x - x_1) $$
where $(x_1, y_1)$ is the point (1, 1) and $m$ is the slope we just found ($\frac{1}{4}$):
$$ y - 1 = \frac{1}{4}(x - 1) $$
- Simplify to slope-intercept form
Now we simplify the equation:
$$ y - 1 = \frac{1}{4}x - \frac{1}{4} $$
$$ y = \frac{1}{4}x + \frac{3}{4} $$
Thus, the equation of the line that is perpendicular to the given line and passes through the point (1, 1) in slope-intercept form is:
$$ y = \frac{1}{4}x + \frac{3}{4} $$
The equation of the line is given by:
$$ y = \frac{1}{4}x + \frac{3}{4} $$
More Information
The obtained equation represents a line that has a slope of $\frac{1}{4}$ and intercepts the y-axis at $\frac{3}{4}$. This line is perpendicular to the original line given by $4x + y = 2$.
Tips
- A common mistake is miscalculating the negative reciprocal of the slope. Always ensure that you change the sign and flip the fraction.
- Another mistake could be in the algebra when rearranging or simplifying the point-slope form. Double-check each step for arithmetic errors.
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