# Find the equation of the straight line passing through the point of intersection of 2x+y-1=0 and x+3y-2=0 and making with the coordinate axes a triangle of area 3/8

1
by 2092000

2015-03-22T03:16:08+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
The point of intersection of     2 x + y = 1  ---  (1)        and       x + 3 y = 2    -- (2)    is  found by   (1) * 3 -  (2) .
6x + 3 y - x - 3y = 3 - 2
=>  x = 1/5    and     y = 3/5  using (1)
Point of intersection is A (1/5,  3/5)

See the diagram attached.

Equation of a line with an x intercept of a and y intercept of b is:
x / a + y / b  = 1          ---  (3)
As the point A lies on the above straight line,
1/(5a)  + 3/(5b) = 1      --- (4)

Area of the triangle formed is = 1/2 * | a b |  = 3/8    --- (5)
So  either  a* b  = 3/4      =>    b = 3/(4a)    --- (6)
or    a * b  = - 3/4    =>  b = -3/(4a)    --- (7)

We use equation (4) and  (6) together:
1/(5a) + 3/5 * 4a/3 = 1
1/a + 4a = 5        =>  4 a² - 5 a + 1 = 0
=>  (4a - 1) (a -1) = 0  =>  a = 1 or  1/4
=>  b = 3/(4a) = 3/4  or  3

So the equations of two possible lines are:
x + 4/3 y = 1    =>   3x + 4y = 3           --  line A in the diagram.
4 x + y/3 = 1    =>  12 x + y = 3          -- line B in the diagram.
=======
Now, we use the equations (4) and (7) together:
1/(5a) - 3/5 * 4a/3 = 1
1/a - 4a = 1
4 a² + a - 1 = 0      =>  a  = [ -1 +- √(1+16) ] /8
a = (√17 - 1)/8      or      -(√17 + 1)/8
=>      b = -6/(√17-1)        or      6/(√17+1)

Hence the equations of these two straight lines are :
8 x /(√17-1)  - (√17-1) y/6 = 1  =>  24 x - (9 - √17) y = 3(√17 -1)          -- Line C
- 8x/(√17 + 1) + (√17+1) y/6 = 1  => - 24 x + (9+√17) y = 3(√17+1)      -- Line D.

So there are 4 lines.  Two in the 1st quadrant, one in the second and one in the fourth quadrant.