Chapter 0.4, Problem 3QR

To determine

To write: An equations for the line passing through a point.

Answer to Problem 3QR

The required equation is of the line is $y=2$ .

Explanation of Solution

Given data: The vertical line passes through point $\left(2,-3\right)$ .

Formula used:

A line having slope m and passing through a point $\left(h,k\right)$ is given by

  $y-k=m\left(x-h\right)$ … (1)

The slope m of a line that makes angle $\theta$ with x -axis is given by $m=\tan \theta$ .

Calculation:

The line is vertical and passes through the point $\left(2,-3\right)$ .

Since the line is vertical, i.e., parallel to y -axis that makes angle ${90}^{\text{ο}}$ with x-axis, therefore, its slope m is

  $m=\tan {90}^{\text{ο}}=\infty$

Substitute $\infty$ for m , 2 for h and -3 for k in equation (1),

  $\begin{array}{c}y-\left(-3\right)=\infty \left(x-2\right)\\ y+3=\infty \left(x-2\right)\\ \frac{y+3}{\infty }=x-2\\ 0=x-2\because \frac{a}{\infty }=0\\ x=2\end{array}$

Therefore, the equation of the vertical line that passes through the point $\left(2,-3\right)$ is $x=2$ .

Direct Method: The equation of a line parallel to y axis and passing through point $\left(h,k\right)$ is given by

  $x=h$ .Therefore, the equation of the vertical line that passes through the point $\left(2,-3\right)$ is $x=2$ .