Chapter 3.4, Problem 57HP

To determine

Write equations of two perpendicular lines that intersect at the point (-3,-7).

Answer to Problem 57HP

The equations of two lines that are perpendicular and intersect at (-3,-7) are :

  $y=x-4$

  $y=-x-10$

Explanation of Solution

Formula Used:

• Point-slope form of a line passing through $(x_0,y_0)$ and have slope m : $(y-y_0)=m(x-x_0)$
• The product of slopes of two perpendicular lines is -1.

Calculation :

Given that the two lines are perpendicular and passes through (-3,-7).

Let the slope of lines be $m_1\text{ and }{m}_2$

Since the lines are perpendicular, the product of their slope is -1.

  $m_1\cdot m_2=-1$

We need to find any lines.

Let the slope of one line = $m_1=1$

So,

  $\begin{array}{l}{m}_1\cdot m_2=-1\\ \Rightarrow (1)\cdot m_2=-1\\ \Rightarrow m_2=-1\end{array}$

So, equation of first line passing through (-3,-7) having slope 1:

  $\begin{array}{l}\left(y-(-7)\right)=1\left(x-(-3)\right)\\ \Rightarrow y+7=x+3\\ \Rightarrow y=x+3-\mathrm{7.................}[\text{subtract 7 from each side]}\\ \Rightarrow y=x-4\end{array}$

Equation of second line passing through (-3,-7) having slope -1:

  $\begin{array}{l}\left(y-(-7)\right)=-1\left(x-(-3)\right)\\ \Rightarrow y+7=-x-3\\ \Rightarrow y=-x-3-\mathrm{7.................}[\text{subtract 7 from each side]}\\ \Rightarrow y=-x-10\end{array}$

So, the equations of two lines that are perpendicular and intersect at (-3,-7) are :

  $y=x-4$

  $y=-x-10$