Question

Find the equation of a line perpendicular to y = - 1/4 x + 9 that passes through the point (4, - 8) .

Answer 1

perpendicularWe were given the following information:

The equation of a line is given by: y = - 1/4 x + 9

We want to obtain the equation for a line perpendicular to this line & that passes through the point (4, -8). This is shown below:

The general equation of a straight line is given by:

`\begin{gathered} y=mx+b \\ where\colon \\ m=slope \\ b=y-intercept \end{gathered}`

The equation of the line given us is:

`\begin{gathered} y=-(1)/(4)x+9 \\ \text{Comparing this with the general equation, we will deduce that:} \\ mx=-(1)/(4)x \\ m=-(1)/(4) \\ \text{Thus, the slope of this line is: }-(1)/(4) \end{gathered}`

The relationship between the slope of a line and the slope of a line perpendicular to it is given by the statement "the product of the slopes of two lines perpendicular to one another is negative one"

This is expressed below:

`\begin{gathered} m* m_(perpendicular)=-1 \\ m_(perpendicular)=-(1)/(m) \\ m=-(1)/(4) \\ m_(perpendicular)=(-1)/(-((1)/(4))) \\ m_(perpendicular)=4 \\ \\ \therefore m_(perpendicular)=4 \end{gathered}`

Therefore, the slope of the perpendicular line is: 4

We were told that the perpendicular line passes through the point (4, -8). We will obtain the equation of the perpendicular line using the Point-Slope equation. This is shown below:

`\begin{gathered} y-y_1=m(x-x_1) \\ (x_1,y_1)=(4,-8) \\ m\Rightarrow m_(perpendicular)=4_{} \\ \text{Substitute the values of the variables into the initial equation above, we have:} \\ y-\mleft(-8\mright)=4(x-4) \\ y+8=4(x-4) \\ y+8=4x-16 \\ \text{Subtract ''8'' from both sides, we have:} \\ y=4x-16-8 \\ y=4x-24 \\ \\ \therefore y=4x-24 \end{gathered}`

The graphical representation of this is given below: