Question

How do you write an equation of a line with an undefined slope and passing through the point $\left( { - 2,4} \right)$ ?

Answer 1

Hint: We need to write an equation of a line with an undefined slope. So, first of all, if the slope of a line is undefined, then this means that ${x_2} - {x_1}$ in the slope of a line is equal to zero. This means that in this case, a slope cannot be obtained, as the denominator of the fraction becomes zero. Using this information, we will write the equation of a line.

Formula used: The slope of a line passes through points $({x_1},{y_1})$ and $({x_2},{y_2})$ : $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{\Delta y}}{{\Delta x}}$ .

Complete step-by-step solution:
As we know we can determine the slope of a line that passes through two points by using the formula of the slope of a line.
In other words, we find the difference between the two values of $y$ and divide the new value by the difference between the two values of $x$ , as mentioned in the formula above.
Now, a line has an undefined slope means that the value of the slope is infinite or unattainable. This is when the denominator of a fraction is equal to zero. Also, we know that we cannot divide by zero. So, in this case, the slope is undefined.
So, we take the denominator from the formula above and put it equal to zero to ensure that we will achieve an undefined slope.
Now, we get, ${x_2} - {x_1} = \Delta x = 0$
We can put $- 2$ for ${x_1}$ or ${x_2}$ . So, we get
${x_2} - ( - 2) = 0$
$\Rightarrow {x_2} + 2 = 0$
Further, we solve it for ${x_2}$ .
$\Rightarrow {x_2} = - 2$
This shows that as long as $x$ is equal to $- 2$ , the line will have an undefined slope.

Therefore, the required equation of line is $x = - 2$.

Note: Since the line has an undefined slope. This indicates that it is a vertical line parallel to the $y - axis$ and passing through all points in the plane with the same x-coordinate. Here, the line passes through $( - 2,4)$ . It will also pass through the points $( - 2,0)$ , $( - 2, - 4)$ and so on.