coordinate geometry sum

Question

find the equation of the line perpendicular to the given line X+4y-8=0 which passes through the point (2,8) in standard form

Answer 1

There are some very nice characteristics or the equation of a line

in Standard Form (Ax + By = C):

   (1) It may represent a vertical line  (for example x=2)

   (2) The slope is -A/B

   (3) The slope of all parallel lines is B/A (negative reciprocal); their equation is:  By - Ax = D

            (use letters, convert to slope-intercept form and verify this)

   (4) The y-intercept is C/A   [the value of y when x=0]

   (5) The x-intercept is C/B   [the value of x when y=0]

Once you learn these, you won't have to be continually converting to slope-intercept form.

 

The problem gives the formula as:  x+4y-8=0, which in Standard Form is:

     x + 4y = 8

All perpendicular lines have the formula:  4x - y = D    (where D is another constant, not C)

 

The point (2,8) will give us the value of D:

    4(2) - (8) = D

      0  = D

 

The equation of the perpendicular line, in Standard Form, is:

     4x - y = 0

 

Checking (very important):

  Does 4x-y=0 contain the point (2,8)  ?

          8 - 8 = 0   ?   yes

Answer 2

To find the equation of a perpendicular line, the first thing you need is the slope.  The most straightforward way to see slope is to turn your equation into a slope-intercept form equation (y=mx+b) and use the "m".

 

x+4y-8=0  isolate the y term

4y=-x+8    divide by 4

y=-¼x+2

 

So the original slope is -¼.  The perpendicular line has a negative reciprocal slope, so that will be 4.

 

To find the equation of a line when you have a point that is not the intercept, the best place to start is with the point-slope form of the equation which is y-y₁=m(x-x₁).  m=4, x₁=2, y₁=8

 

y-8=4(x-2)

y-8=4x-8

 

To make this standard form, move the y to the x side and the constants to the other side.

-8+8=4x-y

4x-y=0  is the standard form of the line.