A line passes through the points A
Question:

A line passes through the points A(4, -6) and B(-2, -5). Show that the line AB makes an obtuse angle with the x-axis.

Solution:

For the line to make an obtuse angle with X-axis, the angle of the line should be greater than 90

For the angle to be greater than $90^{\circ}$, $\tan \theta$ must be negative

Where tanθ is the slope of the line.

Given points are A(4, -6) and B(-2, -5)

slope $=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)$

The slope of line $A B$ is $\left(\frac{-5+6}{-2-4}\right)=\frac{1}{-6}=\frac{-1}{6}$

Which is less than 0, hence negative

$\Rightarrow \tan \theta=\frac{-1}{6}\left\langle 0, \tan \theta\right.$ is negative in $2^{\text {nd }}$ quadrant whose angle is $>90^{\circ} .$

So line $A B$ makes obtuse angle $(>90)$ with the $X$-axis.