Question

The angle between the line $x + y = 3$ and the line joining the points $(1, 1)$ and $(-3, 4)$ is:

• A. $\tan^{-1}(7)$
• B. $\tan^{-1}\left(-\frac{1}{7}\right)$
• C. $\tan^{-1}\left(\frac{1}{7}\right)$
• D. $\tan^{-1}\left(\frac{2}{7}\right)$

Answer 1

The Correct Option is C

1. Understand the problem:

We need to find the angle between the line \( x + y = 3 \) and the line joining the points (1, 1) and (-3, 4).

2. Find the slope of the given line \( x + y = 3 \):

Rewrite the equation in slope-intercept form:

\[ y = -x + 3 \implies m_1 = -1 \]

3. Find the slope of the line joining (1, 1) and (-3, 4):

The slope \( m_2 \) is:

\[ m_2 = \frac{4 - 1}{-3 - 1} = \frac{3}{-4} = -\frac{3}{4} \]

4. Compute the angle \( \theta \) between the two lines:

The formula for the angle between two lines with slopes \( m_1 \) and \( m_2 \) is:

\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]

Substituting \( m_1 = -1 \) and \( m_2 = -\frac{3}{4} \):

\[ \tan \theta = \left| \frac{-1 - \left(-\frac{3}{4}\right)}{1 + (-1)\left(-\frac{3}{4}\right)} \right| = \left| \frac{-\frac{1}{4}}{1 + \frac{3}{4}} \right| = \left| \frac{-\frac{1}{4}}{\frac{7}{4}} \right| = \frac{1}{7} \]

Thus, \( \theta = \tan^{-1} \left( \frac{1}{7} \right) \).

Correct Answer: (C) \(\tan^{-1} \left( \frac{1}{7} \right)\)

Answer 2

First, let's find the slope of the line \( x + y = 3 \). We can rewrite this in slope-intercept form (\( y = mx + b \)):

\[ y = -x + 3 \]

The slope of this line (\( m_1 \)) is \( -1 \).

Next, let's find the slope of the line joining the points \( (1, 1) \) and \( (-3, 4) \). The slope formula is:

\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 1}{-3 - 1} = \frac{3}{-4} = -\frac{3}{4} \]

Now we need to find the angle between these two lines. The formula for the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is:

\[ \tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

Substituting the slopes we found:

\[ \tan \theta = \left| \frac{-\frac{3}{4} - (-1)}{1 + (-1)(-\frac{3}{4})} \right| = \left| \frac{\frac{1}{4}}{\frac{7}{4}} \right| = \frac{1}{7} \]

Therefore, the angle between the lines is:

\[ \theta = \tan^{-1}\left(\frac{1}{7}\right) \]