Q78 / 180 · COMEDK 2023 (Evening Shift) · Mathematics
What can be said regarding a line if its slope is negative?
- Correct answerA$\theta$ is an obtuse angle
- B$\theta$ is equal to zero
- C$\theta$ is an acute angle
- DEither the line is $x$ axis or it is parallel to the $x$ axis
Solution
The slope $m$ of a line is defined as $m = \tan \theta$, where $\theta$ is the angle of inclination of the line with the positive direction of the $x$-axis, such that $0 \le \theta < \pi$.
Given that the slope $m$ is negative, we have $\tan \theta < 0$.
In the interval $[0, \pi)$, the tangent function is negative only when $\theta$ lies in the second quadrant, which corresponds to $\dfrac{\pi}{2} < \theta < \pi$.
An angle $\theta$ in the range $\dfrac{\pi}{2} < \theta < \pi$ is by definition an obtuse angle.
Answer: $\theta$ is an obtuse angle
Given that the slope $m$ is negative, we have $\tan \theta < 0$.
In the interval $[0, \pi)$, the tangent function is negative only when $\theta$ lies in the second quadrant, which corresponds to $\dfrac{\pi}{2} < \theta < \pi$.
An angle $\theta$ in the range $\dfrac{\pi}{2} < \theta < \pi$ is by definition an obtuse angle.
Answer: $\theta$ is an obtuse angle
Hence, the correct answer is option: 'A'.