Question

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

Answer 1

Hint: We know that the slope of a line is \[m = \tan \theta \]. By using the formula \[m = \tan \theta \] we will calculate the slope of the line. Then we will put \[m\] and \[c\] in the equation \[y = mx + c\] to get equation required equation.

Formula used:
The slope of a line is \[m = \tan \theta \], where the straight line makes an angle \[\theta \] with x-axis.
The slope intercept form of a line is \[y = mx + c\], where \[m\] is the slope of the line and \[c\] is y-intercept.

Complete step by step solution:
Given that a straight line makes an angle \[{135^ \circ }\] with the x-axis.
By applying the slope formula, we will calculate the slope of the line.
Here \[\theta = {135^ \circ }\].
The slope of the line is \[m = \tan {135^ \circ }\].
\[ = \tan \left( {{{180}^ \circ } - {{45}^ \circ }} \right)\]
\[ = - \tan \left( {{{45}^ \circ }} \right)\] Since \[\tan \left( {{{180}^ \circ } - \theta } \right) = - \tan \theta \]
\[ = - 1\]
Given that, the y-intercept of the line is -5.
So, the value of \[c\] is -5.
Now we will put the value of \[c\] and \[m\] in the slope intercept form of a line.
\[y = - 1 \cdot x - 5\]
Rewrite the above equation
\[x + y + 5 = 0\]

Hence option C is the correct option.

Note: Many students often confused with the formula \[\tan \left( {{{180}^ \circ } + \theta } \right) = \tan \theta \] and \[\tan \left( {{{180}^ \circ } - \theta } \right) = \tan \theta \] . The correct formulas are \[\tan \left( {{{180}^ \circ } + \theta } \right) = \tan \theta \] and \[\tan \left( {{{180}^ \circ } - \theta } \right) = - \tan \theta \].