What is the equation of the line perpendicular to #y=6x # that passes through # (6,-1) #?

2 Answers
Jul 20, 2016

Answer:

Different way of saying the same thing:

#color(blue)(y=-1/6x)#

Explanation:

#color(blue)("General Introduction")#

Given:#" "y=6x#

Compare to the standard equation form of #y=mx+c#
Where #m# is the gradient (slope)

Looking at the question we see that the gradient is 6.
This means that for 1 along on the x-axis the line has gone up 6 on the y-axis.

#color(blue)(y=6x" "->" gradient "->" "m=6)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(white)(.)#

#color(blue)("Building the equation of the perpendicular line")#

#color(brown)("A line perpendicular to "y=mx+c" has gradient "-1/m#

If #m=6" then "-1/m=-1/6#

So the straight line perpendicular to that given has the equation

#color(blue)(y=-1/mx+c" "->" " y=-1/6x+c)#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(white)(.)#
#color(blue)("Determine the value of the constant "c)#

This line passes through the point #(x,y)->(6,-1)#

So by substitution

#y=-1/6x+c" "->" "-1=-1/(cancel(6))(cancel(6))+c#

#=>c=0#

so the equation of the perpendicular line is:

#color(blue)(y=-1/6x)#

Tony B

Jul 20, 2016

Answer:

#y = -1/6x#

Explanation:

When lines are perpendicular, one slope is the negative reciprocal of the other.
In # y = 6x# the gradient is 6.

The slope perpendicular to this is #-1/6#

We now have the slope and the point #(6,-1)#

There is really nifty formula which applies in a case like this. We have the slope and one point and need to find the equation of the line.

#(y- y_1) = m(x-x_1)# where the given point is #(x_1,y_1)#

Substitute the values given.

#y-(-1) = -1/6(x-6)" "# multiply out and simplify.

#y +1 = -1/6x +1#

#y = -1/6x#