# 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

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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.

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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.**

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

=======

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 =>

- 8x/(√17 + 1) + (√17+1) y/6 = 1 =>