Question

A straight line makes an angle of $135{}^\circ $ with x-axis and cuts y-axis at a distance of $-5$ from the origin. The equation of the line is
A. $2x+y+5=0$
B. $x+2y+3=0$
C. $x+y+5=0$
D. $x+y+3=0$

Answer 1

Hint: In this question, we are to find the equation of the line which makes an angle of $135{}^\circ $ with the x-axis and having an intercept at the y-axis. So, to find this, the slope-intercept form of the straight line is applied.

Formula used: The equation of a line that makes an angle $\theta $ with the x-axis and cuts the y-axis at a distance of $c$ units from the origin is
$y=mx+c$
Where $m=\tan \theta $ and $c$ is the y-intercept.
This equation is also called the slope-intercept form of a line.
If the equation of the line is in the form of $ax+by+c=0$ then the slope is calculated by $m=\dfrac{-a}{b}$ and y-intercept is calculated by $\dfrac{-c}{b}$ and the equation is rewritten in the slope-intercept form.

Complete step by step solution: Given that,
A straight line makes an angle of $135{}^\circ $ with the x-axis.
So, the slope of the line is
$\begin{align}
  & m=\tan \theta \\
 & \text{ }=\tan 135{}^\circ \\
 & \text{ }=-1 \\
\end{align}$
It is also given that; the required straight line cuts the y-axis at a distance of $-5$ from the origin.
So, the y-intercept of the line is
$c=-5$
Then, on substituting these values in the standard form, we get
$\begin{align}
  & y=mx+c \\
 & \Rightarrow y=(-1)x-5 \\
 & \Rightarrow x+y+5=0 \\
\end{align}$

Thus, Option (C) is correct.

Note: Here we need to observe that, the given values are an angle and an intercept. So, we can able to find the slope and y-intercept. Then, by substituting these values, we get the required equation of the line.