Chapter 9.1, Problem 75STP

To determine

To Calculalte:

 Which of the following is an equation for the line that passes through (2, -5) and is perpendicular to $2x+4y=8$ .

Answer to Problem 75STP

Correct option is (C)

Explanation of Solution

Given Information:

Options:

  $\begin{array}{l}A)y=2x+10\\ B)y=-\frac{1}{2}x-4\\ C)y=2x-9\\ D)y=-2x-1\end{array}$

Formula Used:

• Two lines are perpendicular , then the product of their slopes is -1.That is , if the slope of a line is m , then the slope of line perpendicular to it will be $-\frac{1}{m}$ .
• The slope-intercept form of a line is :

  $\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\end{array}$

Calculation:

Since the required line passes through point (2,-5) and is perpendicular to the line $2x+4y=8$ .

1. Finding the slope of required line :
• Convert the given equation $2x+4y=8$ in slope-intercept form :

  $\begin{array}{l}2x+4y=8\\ 2x+4y-2x=8-2x\mathrm{.................}[subtract"2x"each\_side]\\ 4y=8-2x\\ \frac{4y}{4}=\frac{-2x}{4}+\frac{8}{4}\mathrm{...........................}[divide"4"each\_side]\\ y=-\frac{1}{2}x+2\end{array}$

• Comparing the equation $y=-\frac{1}{2}x+2$ with slope-intercept form of a line

  $\begin{array}{l}y=mx+b\\ where\\ m=slope\\ b=y-\mathrm{int}ercept\\ y=-\frac{1}{2}x+2\\ \Rightarrow m=-\frac{1}{2}\\ \&\\ b=y-\mathrm{int}ercept=2\end{array}$

• Since the product of slopes of perpendicular lines is -1 .
• So, the slope of required line is $2$ .
2. Find the y-intercept :

• From the slope-intercept form of a line , since the required line passes through (2,-5) and has slope = 2 .
• Substituting the values and finding the value of b :

  $\begin{array}{l}y=mx+b\\ -5=2(2)+b\mathrm{............}[m=2;(x,y)=(2,-5)]\\ -5=4+b\\ -5-4=4+b-\mathrm{4.....}[subtract"4"each\_side]\\ b=-9\end{array}$

So, the b = -9 and m = 2. Hence, the required equation is :

  $\begin{array}{l}y=mx+b\\ y=2x-\mathrm{9..................}[m=2,b=-9]\end{array}$

So, option C) is correct.