Question

Let $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ be two points on a non-vertical line $l$ whose inclination is $θ$. Then,
I. Slope of the line $l$ is $m=\frac{y_2−y_1}{x_2−x_1}$, whether $θ$ is acute or obtuse.
II. Slope of the line $l$ is $m=\frac{y_2−y_1}{x_2−x_1}$, when $θ$ is acute only.

• A. I and II are true
• B. I is true and II is false
• C. I is false and II is true
• D. I and II are false

Answer 1

Answer: (B)

Case I When angle $θ$ is acute.

From figure, $âˆMPQ=θ$
Therefore, slope of line $I=m=\tan θ....$(i)
But in $â–³MPQ$, we have
$\tan θ=\frac{MQ}{MP}=\frac{y_2−y_1}{x_2−x_1}....$(ii)
From Eqs. (i) and (ii), we have
$m=\frac{y_2−y_1}{x_2−x_1}$
Case II When angle $θ$ is obtuse.

From figure, we have
$âˆMPQ=180^{∘}−θ$
Therefore, $θ=180^{∘}−âˆMPQ$
Now, slope of the line $l$ is
$m=\tan θ$
$=\tan \left(180^{∘}−âˆMPQ\right)$
$=−\tan âˆMPQ$
$=−\frac{MQ}{MP}$
$=−\frac{y_2−y_1}{x_1−x_2}$
$=\frac{y_2−y_1}{x_2−x_1}$
Consequently, we see that in both the cases, the slope $m$ of the line / through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by
$m=\frac{y_2−y_1}{x_2−x_1}\text{. }$