Question

Write the equation of x-axis and y-axis. Also write the equation of lines parallel to x-axis and y-axis.

Answer 1

Hint: In this question, we have to mainly focus on the slopes of the lines for which we are trying to find the equation.

Complete step-by-step solution -
Moving on to the solution, let us first try to get the equation of the x-axis.
We know that the slope is defined as $\text{ }m=\tan \theta $ , where $\theta $ is the angle made by the line with positive x-axis in an anti-clockwise direction.
So, for x-axis $\theta ={{0}^{\circ}}$ hence, making the slope of x-axis:
$m=\tan \theta =\tan {{0}^{
\circ}}=0$
We know, the general equation of the line can be written as:
$y=mx+c$
So, the equation of x-axis becomes:
$y=0.x+c$
$\Rightarrow y=0+c$
$\Rightarrow y=c...........(i)$
Here, c is a constant and to eliminate this, we need a point lying on x-axis and we know x-axis passes through the origin so the point on the line is O (0, 0).
Putting in the equation $\text{ }(i)$ , we get
$y=c$
$\Rightarrow 0=c$
$\therefore c=0$
Therefore, the equation of the x-axis comes out to be $y=0$ .
Similarly, for the y-axis, $\theta ={{\dfrac{\pi }{2}}} $ . Hence, we get the slope of y-axis to be:
\[m=\tan \theta =\tan {{\dfrac{\pi }{2}}}=\infty \]
But we know, $\infty =\dfrac{1}{0}$ . So, our slope becomes:
$m=\infty =\dfrac{1}{0}$
Again, using general equation of the line;
$y=mx+c$
So, the equation of y-axis becomes;
$y=\dfrac{1}{0}.x+c$
$\Rightarrow y-c=\dfrac{1}{0}.x$
On cross-multiplication, we get:
$0.(y-c)=x$
And any finite term multiplied by 0 gives 0.
Now, taking c to be finite, the equation of y-axis comes out to be $x=0$ .
When c is infinite:
We know, infinity multiplied by zero can give any value. So, let the value be ${{c}_{1}}$ .
We get;
$x={{c}_{1}}............(ii)$
Here, ${{c}_{1}}$ is a constant and to eliminate this, we need a point lying on y-axis and we know y-axis passes through the i.e. O (0, 0) is the point lying on the line.
Putting in the equation $\text{ }(ii)$ , we get
$x={{c}_{1}}$
$\Rightarrow 0={{c}_{1}}$
$\therefore {{c}_{1}}=0$
Equation of y-axis comes out to be $x=0$ in both cases, i.e. when c is finite and when c is infinite.
Moving on to lines parallel to x-axis and y-axis.
And we know, two lines are said to be parallel if they have absolute value of their slopes equal.
First, try to find the general equations of lines which are parallel to x-axis.
We know the slope of the line=slope of x-axis=0.
Using general form of line, we get
$y=mx+c$
So, the equation of line becomes;
$y=0.x+c$
$\Rightarrow y=0+c$
$\Rightarrow y=c$
Hence, the equation of line parallel to x-axis is given by $y=c$ .
Moving on to the line parallel to y-axis.
We know the slope of the line parallel to y-axis = slope of y-axis = $\dfrac{1}{0}$ .
Using general form of line, we get
$y=mx+c$
So, the equation of line becomes;
$y=\dfrac{1}{0}.x+c$
$\Rightarrow y-c=\dfrac{1}{0}.x$
On cross-multiplication, we get:
$0.(y-c)=x$
When c is finite;
The equation of the line parallel to y-axis come out to be;
$x=0$
When c is infinite:
We know, infinity multiplied by zero can give any value so let the value be ${{c}_{1}}$ .
We get, $x={{c}_{1}}$ .
As 0 is also a constant, the equation of line parallel to y-axis comes out to be: $x=k$
Where k is constant.

Note: Alternately, for finding the equation of x-axis and y-axis you can assume two points lying on them and solve them directly. For instance points on the x axis can be O (0, 0) and A (1, 0).
Also avoid using intercept form of line for the lines parallel to x-axis and y-axis as you may get one of the intercepts to be infinity.