Question

What can be said regarding a line if its slope is
i) Zero
ii) Positive
iii) Negative
iv) Not defined

Answer 1

Hint: Use the fact that the slope of the line can also be represented as the tangent of the angle which the line makes with the positive x-axis when going anticlockwise from the x-axis.
 The value of m gives the slope of the line and then equate it to the tangent of the angle which the line makes with the positive x-axis when going anticlockwise from the x-axis as follows
\[m=\tan \theta \]
(Where \[\theta \] is the angle that the line makes with the positive x-axis when going anticlockwise from the x-axis and m is the slope of the line which is inclined to the x-axis with the mentioned angle)
Complete step-by-step answer:
Now, in this question, try to get a generic answer for each of the options by looking at the value that we will get through the given formula.
As mentioned in the question, we have to find what can be said about the slopes of the lines with the given slopes which are mentioned in the question.
i) Zero
Now, on using the formula that is given in the hint, we can write the following
\[\begin{align}
  & m=\tan \theta \\
 & m=0=\tan \theta \\
 & \theta ={{\tan }^{-1}}0 \\
 & \theta =0 \\
\end{align}\]
Hence, we can say that the line with zero as the slope is parallel to the x-axis.
ii) Positive
Now, on using the formula that is given in the hint, we can write the following
\[\begin{align}
  & m=\tan \theta \\
 & m=+ve=\tan \theta \\
\end{align}\]
Now, we know that the value of a tan function’s output is positive only when the angle on which it is being applied upon is an acute angle.
Hence, we can say that the line with positive slope makes an acute angle with the x-axis when measured anti-clockwise.
 iii) Negative
Now, on using the formula that is given in the hint, we can write the following
\[\begin{align}
  & m=\tan \theta \\
 & m=-ve=\tan \theta \\
\end{align}\]
Now, we know that the value of a tan function’s output is negative only when the angle on which it is being applied upon is an obtuse angle.
Hence, we can say that the line with negative slope makes an obtuse angle with the x-axis when measured anti-clockwise.
iv) Not defined
Now, on using the formula that is given in the hint, we can write the following
\[\begin{align}
  & m=\tan \theta \\
 & m=\infty =\tan \theta \\
 & \theta ={{\tan }^{-1}}\infty \\
 & \theta ={{90}^{\circ }} \\
\end{align}\]
Hence, we can say that the line with not defined slope is parallel to the y-axis and is perpendicular to the x-axis, that is it makes \[{{90}^{\circ }}\] with the x-axis.

Note: The students can make an error if they don’t know about the formulae that are given in the hint as without knowing them one can never get to the correct answer.
We can also look at the diagrams to get a better picture which is as follows
(i)

(ii)

(iii)

(iv)

Also, it is important to know the basic values and the basic properties of tan function for solving this question as without knowing them one can never get to the correct answer.