Question

What is the equation of a horizontal line passing through \[\left( { - 3, - 5} \right)\] \[?\]

Answer 1

Hint: We have to find the equation of a horizontal line passing through \[\left( { - 3, - 5} \right)\]. Every horizontal line is parallel to the x-axis i.e., \[y = 0\]. Hence a horizontal line passing through a point \[\left( { - 3, - 5} \right)\] is parallel to the x-axis. Using slope-point form of the equation, find the equation of the horizontal line passing through a point \[\left( { - 3, - 5} \right)\].

Complete step by step solution:
Parallel lines: Let \[{L_1}\] and \[{L_2}\] be two parallel lines with the slopes \[{m_1}\]and \[{m_2}\]respectively. then\[{m_1} = {m_2}\].
Perpendicular lines: Let \[{L_1}\] and \[{L_2}\] be two perpendicular lines with the slopes \[{m_1}\]and \[{m_2}\]respectively. then \[{m_1} \times {m_2} = - 1\].
Slope-point form: The equation of the straight line passing through the point \[\left( {a,b} \right)\] and with slope \[m\]is given by \[(y - b) = m(x - a)\].
Given a line passing through a point \[\left( { - 3, - 5} \right)\] is parallel to the x-axis. Suppose \[{m_1}\]and \[{m_2}\] be the slopes of the horizontal line and x-axis respectively. Then from the equation of the x-axis \[y = 0\]\[ \Rightarrow {m_2} = 0\]
Since \[{m_1} = {m_2}\]\[ \Rightarrow {m_1} = 0\].
By Slope-point form, the equation of the line passing through the point\[\left( { - 3, - 5} \right)\] and parallel to the x-axis is
\[y - ( - 5) = 0(x - ( - 3))\]
\[ \Rightarrow y = - 5\].
 Hence, the equation of a horizontal line passing through \[\left( { - 3, - 5} \right)\] is \[y = - 5\].
So, the correct answer is “\[y = - 5\].”.

Note: Note that the vertical lines in the coordinate system are perpendicular to the y-axis i.e., \[x = 0\].
The general form of the equation of the straight line is \[ax + by + c = 0\].