Chapter 3.4, Problem 37WE

(a)

To determine

To Find: To find an equation passing through a point and parallel to the given line.

Answer to Problem 37WE

The required equation is $y-1=0$

Explanation of Solution

Given information: Given line $y+2=0\text{and}\text{point}P(-4,1)$

Formula used: For parallel of lines $m_1=m_2$ and for equation of line $y-y_1=m(x-x_1)$

Calculation:

  $\begin{array}{l}\text{Equation}\text{of}\text{the}\text{given}\text{line}y+2=0\\ \text{converting}\text{the}\text{line}\text{into}\text{slope-intercept}\text{form}\\ y=-2\\ y=0x-2\\ \text{comparing}\text{with}y=mx+c,\text{we}\text{get}\\ \text{slope}(m)=0\\ \because \text{lines}\text{are}\text{parallel,}\\ \therefore \text{Slope}\text{of}\text{the}\text{required}\text{line}=\text{Slope}\text{of}\text{the}\text{given}\text{line}\\ \text{So,}\text{Slope}\text{of}\text{the}\text{required}\text{line}\text{is}\text{also}0.\\ \text{Now,}\text{equation}\text{of}\text{the}\text{required}\text{line,}\text{using}\text{point-slope}\text{form}\\ y-1=0(x+4)\\ y-1=0\text{which}\text{is}\text{the}\text{required}\text{equation}\text{in}\text{standard}\text{form}\text{.}\end{array}$

Hence, the equation of line is $y-1=0$ which is parallel to given line.

(b)

To determine

To Find: To find an equation passing through a point and perpendicular to the given line.

Answer to Problem 37WE

The required equation is $x+4=0$

Explanation of Solution

Given information: Given line $y+2=0\text{and}\text{point}P(-4,1)$

Formula used: For perpendicular of lines $m_1\times m_2=-1$ and for equation of line $y-y_1=m(x-x_1)$

Calculation:

  $\begin{array}{l}\text{Equation}\text{of}\text{the}\text{given}\text{line}y+2=0\\ \text{converting}\text{the}\text{line}\text{into}\text{slope-intercept}\text{form}\\ y=-2\\ y=0x-2\\ \text{comparing}\text{with}y=mx+c,\text{we}\text{get}\\ \text{slope}(m)=0\\ \because \text{lines}\text{are}\text{perpendicular,}\\ \therefore \text{Slope}\text{of}\text{the}\text{required}\text{line}\times \text{Slope}\text{of}\text{the}\text{given}\text{line}=-1\\ \text{So,}\text{Slope}\text{of}\text{the}\text{required}\text{line}\text{is}\text{not}\text{defined}\\ \text{Now,}\text{equation}\text{of}\text{the}\text{required}\text{line,}\text{using}\text{point-slope}\text{form}\\ x+4=0\text{which}\text{is}\text{the}\text{required}\text{equation}\text{in}\text{standard}\text{form}\text{.}\end{array}$

Hence, the equation of line is $x+4=0$ which is parallel to the given line.