SOLUTION: PLEASE HELP. WHAT IS THE EQUATION OF A LINE THROUGH (4, -1) AND WHOSE SEGMENT INTERCEPTED BETWEEN THE AXES IN THE 4TH QUADRANT IS EQUAL TO {{{ 2sqrt(17) }}}. THANKS IN ADVANCE.

Algebra ->  Test -> SOLUTION: PLEASE HELP. WHAT IS THE EQUATION OF A LINE THROUGH (4, -1) AND WHOSE SEGMENT INTERCEPTED BETWEEN THE AXES IN THE 4TH QUADRANT IS EQUAL TO {{{ 2sqrt(17) }}}. THANKS IN ADVANCE.      Log On


   

Question 875156: PLEASE HELP. WHAT IS THE EQUATION OF A LINE THROUGH (4, -1) AND WHOSE SEGMENT INTERCEPTED BETWEEN THE AXES IN THE 4TH QUADRANT IS EQUAL TO +2sqrt%2817%29+. THANKS IN ADVANCE.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
DISCLAIMER:
The solution that follows is not elegant.
I would welcome any alternative.
If this strategy was the expected one, and maybe there is no better way, I would love to know that too.

The situation in the problem is illustrated in the drawing below.
We are looking for m and the coordinates of x%5BA%5D and y%5BB%5D of A and B,
and all of them should be negative.
We use y%2B1=m%28x-4%29<--->y=mx-%284m%2B1%29 to find x%5BA%5D and y%5BB%5D as a function of m .
system%28y=mx-%284m%2B1%29%2Cx=0%29 ---> system%28x=0%2Cy=4m%2B1%29 ---> y%5BB%5D=4m%2B1
system%28y=mx-%284m%2B1%29%2Cy=0%29 ---> system%28y=0%2Cx=%284m%2B1%29%2Fm%29 ---> x%5BA%5D=%284m%2B1%29%2Fm
Since AB=2sqrt%2817%29<-->AB%5E2=2%5E2%2A17<-->AB%5E2=68
and AB%5E2=x%5BA%5D%5E2%2By%5BB%5D%5E2 ,
x%5BA%5D%5E2%2By%5BB%5D%5E2=68
Substituting x%5BA%5D=%284m%2B1%29%2Fm and y%5BB%5D=4m%2B1 we get
%28%284m%2B1%29%2Fm%29%5E2%2B%284m%2B1%29%5E2=68
%284m%2B1%29%5E2%2Fm%5E2%2B%284m%2B1%29%5E2=68
%284m%2B1%29%5E2%2Bm%5E2%284m%2B1%29%5E2=68m%5E2
%284m%2B1%29%5E2%281%2Bm%5E2%29=68m%5E2
%2816m%5E2%2B8m%2B1%29%281%2Bm%5E2%29=68m%5E2
%2816m%5E2%2B8m%2B1%29%281%2Bm%5E2%29=68m%5E2
16m%5E4%2B8m%5E3%2B17m%5E2-68m%5E2%2B8m%2B1=0
16m%5E4%2B8m%5E3-51m%5E2%2B8m%2B1=0
There are 4 real solutions to that equation, and we can find their approximate values. I suppose that using a graphing calculator was expected, but I used the computer, since I am already using it to access the answer this question.
The solution we are interested in is the one with
-1%2F4%3Cm%3C0 ,
because that would give us system%28x%5BA%5D%3C0%2Cy%5BB%5D%3C0%29 .
That solution is highlight%28m=-0.08184155%29 ,
which makes 4m%2B1=0.6726338 ,
so the equation of the line is
highlight%28y=-0.08184155x-0.6726338%29