To find the equations of the lines passing through the point [tex]\((2,3)\)[/tex] and making an angle of [tex]\(45^\circ\)[/tex] with the line [tex]\(x - 3y - 2 = 0\)[/tex], follow these steps:
1. Convert the given line to slope-intercept form:
The given line is [tex]\(x - 3y - 2 = 0\)[/tex]. To find the slope, rearrange it to the slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[
x - 3y - 2 = 0 \implies 3y = x - 2 \implies y = \frac{1}{3}x - \frac{2}{3}
\][/tex]
Therefore, the slope ([tex]\(m_1\)[/tex]) of the given line is [tex]\(\frac{1}{3}\)[/tex].
2. Determine the slopes of the lines making [tex]\(45^\circ\)[/tex] with the original slope:
The slopes of the lines making an angle [tex]\(\theta = 45^\circ\)[/tex] with a line of slope [tex]\(m_1\)[/tex] can be found using the formulas:
[tex]\[
m_2 = \frac{m_1 + \tan(\theta)}{1 - m_1 \tan(\theta)}
\][/tex]
[tex]\[
m_3 = \frac{m_1 - \tan(\theta)}{1 + m_1 \tan(\theta)}
\][/tex]
Given [tex]\(\theta = 45^\circ\)[/tex], [tex]\(\tan(45^\circ) = 1\)[/tex].
Thus, the slopes [tex]\(m_2\)[/tex] and [tex]\(m_3\)[/tex] are calculated as:
[tex]\[
m_2 = \frac{\frac{1}{3} + 1}{1 - \frac{1}{3} \cdot 1} = \frac{\frac{4}{3}}{\frac{2}{3}} = 2
\][/tex]
[tex]\[
m_3 = \frac{\frac{1}{3} - 1}{1 + \frac{1}{3} \cdot 1} = \frac{-\frac{2}{3}}{\frac{4}{3}} = -\frac{1}{2}
\][/tex]
3. Find the y-intercepts of the new lines passing through [tex]\((2,3)\)[/tex]:
Using the point-slope form of the equation of a line, [tex]\(y - y_1 = m(x - x_1)\)[/tex], we can find the y-intercepts ([tex]\(b\)[/tex]) for both lines.
For the line with slope [tex]\(m_2 = 2\)[/tex]:
[tex]\[
y - 3 = 2(x - 2) \implies y - 3 = 2x - 4 \implies y = 2x - 1
\][/tex]
Thus, the y-intercept [tex]\(b_2\)[/tex] is [tex]\(-1\)[/tex].
For the line with slope [tex]\(m_3 = -\frac{1}{2}\)[/tex]:
[tex]\[
y - 3 = -\frac{1}{2}(x - 2) \implies y - 3 = -\frac{1}{2}x + 1 \implies y = -\frac{1}{2}x + 4
\][/tex]
Thus, the y-intercept [tex]\(b_3\)[/tex] is [tex]\(4\)[/tex].
4. Write the equations of the lines:
The equations of the lines passing through [tex]\((2,3)\)[/tex] and making an angle of [tex]\(45^\circ\)[/tex] with the line [tex]\(x - 3y - 2 = 0\)[/tex] are:
[tex]\[
\text{Line 1: } y = 2x - 1
\][/tex]
[tex]\[
\text{Line 2: } y = -\frac{1}{2}x + 4
\][/tex]
To summarize, the equations of the two lines are:
[tex]\[ y = 2x - 1 \][/tex]
[tex]\[ y = -\frac{1}{2}x + 4\][/tex]
