A line through (6,2) forms a triangle with the positive arms of the coordinate axes. If the area of the triangle is 27 units^2, find the equation of the line.

Question

 A line through (6,2) forms a triangle with the positive arms of the coordinate axes. If the area of the triangle is 27 units^2, find the equation of the line.

Answer 1

Hi Loppy,

 

This one is really easier to understand if you have a picture.

 

Draw the x- and y-axis, and make a point at (6,2).   Draw the hypotenuse of the right triangle made up of the x-axis for the horizontal leg, and the y-axis for the vertical leg, and the diagonal you are going to draw through the point (6,2).

 

Where the diagonal line through (6,2) crosses the x-axis, call that point "a".  Where the diagonal line crosses the y-axis call that "b"

 

The slope of the diagonal you drew is (-b)/a  

that is down "b" and to the right "a", so in the equation of the line, "m" = -b/a    and we will have to figure out b and a a differnt way.

 

The area of a triangle is (1/2)(base)(height) and for us the base is "a" and the height is "b".  So A=(1/2)ab=27

So this could be simplified to say that ab=54   (multiply both sides by 2)

 

Using point-slope form, slope=(-b/a) and point = (6,2) we get

 

y-2 = (-b/a)(x-6)

 

This can be rearranged into y=mx+b form:

 

y - 2 = (-b/a)x +6b/a     or

 

 

y = (-b/a)x + 6b/a +2

 

The "6b/a + 2" is the y-intercept, but because of the way we drew it, we know that the y-intercept is "b".  

 

That means that 6b/a + 2 = b

 

If we solve ab=54 for "a", then a = 54/b and we will substitute this in for the "a" in the expression "6b/a + 2"

 

6b/(54/b) + 2 = b

 

Simplify the complex fraction:  6b/(54/b)=6b*(b/54) = 6b²/54 = b²/9

 

So the equation becomes b²/9 +2 = b

 

Multiply EVERYTHING by 9 to get rid of the fraction

 

b²+18=9b

 

Move the "9b" to the left and get

 

b²-9b+18=0

 

This factors into

 

(b-6)(b-3)=0

 

So you have two choices for the value of b.

 

It is either b=6 (so a would =9)   or  b=3 (then a would =18)

 

 

Going back up to the top:   the slope, m=-b/a = -6/9 = -2/3     or    m=-b/a=-3/18 = -1/6

 

For both of these the y-intercept is b =6  or b = 3

 

 

So there are two correct solutions:

 

when b=6: y=(-2/3)x+6

when b = 3:  y=(-1/6)x+3

 

 

 

Check:  both of these lines pass through (6,2).  

 

When b = 6, where does the line cross the x-axis?  

If y=(-2/3)x+6, when y=0 is where it crosses the x-axis so 0=(-2/3)x+6

(2/3)x = 6

x = 9

A=(1/2)xy = (1/2)(9)(6)= 27   Works!

 

When b=3, where does the line cross the x-axis?

If y=(-1/6)x+3, when y=0 is where it crosses the x-axis so 0=(-1/6)x+3

(1/6)x=3

x=18

A=(1/2)xy=(1/2)(18)(3)=27  Works!

 

So you have two correct solutions to the question!