Question

Choose the correct angle between the lines $xy = 0$ from the options given below.

1. 45 degrees
2. 60 degrees
3. 90 degrees
4. 180 degrees

Answer 1

First of all, we need to find out the line equations from the given equation $xy = 0$ and then compare the line equations with the general form of the line equations it is ${a_1}x + {b_1}y + {c_1} = 0$ , and ${a_2}x + {b_2}y + {c_2} = 0$. Then we can know the values of${a_1},{b_1},{c_1},{a_2},{b_2},and{c_2}$ and substitute the values into the angle between the 2 lines formula.

Complete step-by-step solution:
First, we need to find the line equations.
Given the equation is $xy = 0$ we know that if $ab = 0 \Rightarrow a = 0$ or $b = 0$.
Since the given equation is $xy = 0$ implies the line equations are $x = 0,y = 0$.
On comparing the given line equations with the general form of line equations it is ${a_1}x + {b_1}y + {c_1} = 0$ , and ${a_2}x + {b_2}y + {c_2} = 0$. We can get the values of ${a_1},{b_1},{c_1},{a_2},{b_2},and{c_2}$
Therefore, ${a_1} = 1,{b_1} = 0,{c_1} = 0,{a_2} = 0,{b_2} = 1,$and ${c_2} = 0$.
We know the formula of the angle between 2 lines formula. It is
$\tan \theta = \left|{\dfrac{{{a_1}{b_2} - {b_1}{a_2}}}{{{a_1}{a_2} + {b_1}{b_2}}}} \right|$, where $\theta$ is the required angle between two lines.
$\Rightarrow \tan \theta = \left| {\dfrac{{1.1 - 0.0}}{{1.0 + 0.1}}} \right|$
On further simplifying the above equation, we get
$\Rightarrow \tan \theta = \infty$
We know that $\tan \theta$ tends to $\infty$ at $\theta = {90^ \circ }$.
Therefore the angle between the given lines $xy = 0$is nothing but ${90^ \circ }$ or a right angle.
The correct option is 3.

Note: This is one way of solving the problem. Many problems in mathematics can be solved in multiple number of ways. The other way of t=solving this problem is quite easy. The easy method is, observe that the line equations $x = 0,y = 0$ are nothing but the equations of the y-axis, x-axis respectively. We know that the angle between the axes is nothing but ${90^ \circ }$.