SOLUTION: The line (x/a) + (y/b) = 1, where a and b are positive constant, intersects the x- and y-axes at the points A and B respectively. The mid-point of AB lies on the line 2x + y = 10

Question 1200960: The line (x/a) + (y/b) = 1, where a and b are positive constant, intersects the x- and y-axes at
the points A and B respectively. The mid-point of AB lies on the line 2x + y = 10 and
the distance AB = 10. Find the values of a and b.
Answer by Edwin McCravy(20054) (Show Source): You can put this solution on YOUR website!

The line  is in intercept form with x-intercept (a,0) and 
y-intercept (0,b). So the midpoint of AB is  = .

Since it lies on the line , 

,





Since the distance AB = 10











a+2=0;  a-20=0
  a=-2;    a=20

, 
,     
,       

So at first, there seem to be two answers. The first one

a=-2, b=24




with A=(a,0)=(-2,0), B=(0,24) 
and midpoint M of AB =  =  = (-1,12)

Here is solution with a=-2, b=24. The green line is 2x + y = 10.
 

However what seemed like another solution

a=20, b=0

turned out to be extraneous because when we substitute a=20, b=0
in , we get



We have an undefined term in the equation, since division by 0 is
not permitted.

If the equation had first been multiplied through by ab so that we
would have had  then we may have had a second solution, 
but since it was given as , the "second solution" 
was extraneous.  So there is just one solution a=-2, b=24.

Edwin

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