Question

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)`?

Answer 1

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))`