What is the equation of the line that passes through #(6,-4)# and is perpendicular to the line that passes through the following points: #(-2,12),(5,-6) #?

1 Answer
Jan 10, 2018

Answer:

See a solution process below:

Explanation:

First, we need to determine the slope of the line passing through #(-2 , 12)# and #(5, -6)#. The slope can be found by using the formula: #m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))#

Where #m# is the slope and (#color(blue)(x_1, y_1)#) and (#color(red)(x_2, y_2)#) are the two points on the line.

Substituting the values from the points in the problem gives:

#m = (color(red)(-6) - color(blue)(12))/(color(red)(5) - color(blue)(-2)) = (color(red)(-6) - color(blue)(12))/(color(red)(5) + color(blue)(2)) = -18/7#

Let's call the slope of a perpendicular line: #m_p#

Then the rule for find the slope of a perpendicular line is:

#m_p = -1/m#

Substituting gives: #m_p = (-1)/(-18/7) = 7/18#

We can now use the point slope formula to find an equation of the line for the point given in the problem and the slope we calculated. The point-slope form of a linear equation is: #(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))#

Where #(color(blue)(x_1), color(blue)(y_1))# is a point on the line and #color(red)(m)# is the slope.

Substituting again gives:

#(y - color(blue)(-4)) = color(red)(7/18)(x - color(blue)(6))#

Or

#(y + color(blue)(4)) = color(red)(7/18)(x - color(blue)(6))#