Question

Find the line that is perpendicular to y=12x+4
and passes through the point (−3,4)
.

Answer 1

To find the line that is perpendicular to the line given by the equation \( y = 12x + 4 \) and passes through the point \( (-3, 4) \), we need to follow these steps:

1. **Determine the slope of the given line**: The slope of the line \( y = 12x + 4 \) is \( 12 \).

2. **Find the slope of the perpendicular line**: The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \( m \) of the perpendicular line is:
\[
m = -\frac{1}{12}
\]

3. **Use point-slope form to find the equation of the perpendicular line**: The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Plugging in \( m = -\frac{1}{12} \), \( x_1 = -3 \), and \( y_1 = 4 \), we have:
\[
y - 4 = -\frac{1}{12}(x + 3)
\]

4. **Simplify the equation**: Distributing the right-hand side:
\[
y - 4 = -\frac{1}{12}x - \frac{3}{12}
\]
\[
y - 4 = -\frac{1}{12}x - \frac{1}{4}
\]

Now, add \( 4 \) to both sides:
\[
y = -\frac{1}{12}x - \frac{1}{4} + 4
\]

Convert \( 4 \) to a fraction with a common denominator:
\[
4 = \frac{16}{4}
\]
Therefore, we have:
\[
y = -\frac{1}{12}x + \frac{16}{4} - \frac{1}{4} = -\frac{1}{12}x + \frac{15}{4}
\]

5. **Final result**: The equation of the line that is perpendicular to \( y = 12x + 4 \) and passes through the point \( (-3, 4) \) is:
\[
y = -\frac{1}{12}x + \frac{15}{4}
\]