Question

Find the equation of the straight lines drawn through the point $\left( {1, - 2} \right)$, making an angle of ${45^ \circ }$ with the x-axis?

Answer 1

In this question a point and the slope are given. Therefore, we will use the slope form to find the equation. First, we find the intercept with the help of a given point and the slope. Then we will find the equation but there will be two equations of straight lines because the angle can be positive direction as well as negative direction with the x-axis.

Complete step by step answer:
In the above question, we have given a point $\left( {1, - 2} \right)$ and a line passes through it which makes an angle ${45^ \circ }$ with the x-axis.
Therefore, we will use the slope form to find the equation of a line.
Also, there will be two values of slope because the line can make an angle of ${45^ \circ }$ with positive direction as well as negative direction of the x-axis.
Case I:
Point $\left( {1, - 2} \right)$ and slope $= \tan \theta = \tan {45^ \circ } = 1 = m$
Also, we know that $y = mx + c$
Now put the value of y, x, and c in the above equation.
$\Rightarrow - 2 = 1\left( 1 \right) + c$
$\Rightarrow c = - 2 - 1$
$\Rightarrow c = - 3$
Now put the value of c and m in the equation of the line.
$y = x - 3$
Case II:
Point $\left( {1, - 2} \right)$ and slope $= \tan \theta = \tan {-45^ \circ } = - 1 = m$
Also, we know that $y = mx + c$
Now put the value of y, x, and c in the above equation.
$\Rightarrow - 2 = - 1\left( 1 \right) + c$
$\Rightarrow c = - 2 + 1$
$\Rightarrow c = - 1$
Now put the value of $c$ and $m$ in the equation of the line.
$y = - x - 1$
Therefore, the equations of the required straight lines are: $y=x-3$ and $y=-x-1$.

Note:
In this question, if a specific value of slope of a particular direction of line with x-axis is given. Then we have to find only one equation of line. There are many ways to find the equation of a line like we can also use a two-point form.