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 $ax + by + c = 0$.