Question

The variable coefficient p,q,r in the equation of the straight line px+qy+r=0 are connected by the relation pa+qb+rc=0 where a,b and c are fixed contants.show that the variable line always passes through a fixed point?

Answer 1

See below.

Explanation:

Solving

`{(p_1 x + q_1 y + r_1 = 0), (p_2 x + q_2 y + r_2 = 0):}`

for `x,y` we find

#{(x = (q_2 r_1 - q_1 r_2)/(p_2 q_1 - p_1 q_2)), (y = (p_1 r_2-p_2 r_1 )/(p_2 q_1 - p_1 q_2)):}#

which is the intersection point.

Solving now

`{(p_1 a + q_1 a + r_1 c= 0), (p_2a + q_2 b + r_2 c= 0):}`

for `p_1, p_2` we obtain

`{(p_1 = -(b q_1 + c r_1)/a), (p_2 = -(b q_2 + c r_2)/a):}`

substituting now the values found to `p_1,p_2` into the intersection point formulas, we obtain

`{(x=a/c),(y=b/c):}`

so as we can conclude all the lines have a common point with coordinates

`(a/c,b/c)`

Answer 2

Given that the equation `px+qy+r=0` represents different straight lines for different values of the variable coefficient `p,q,r` and those variables are connected by the relation `pa+qb+rc=0` where a,b and c are fixed constants.

We are to show that the variable line always passes through a fixed point.

Given equation of variable straight line `px+qy+r=0 ......[1]`

Given relation `pa+qb+rc=0.....[2]`

Dividing both sides of the given relation by `c` we get the follwing form of relation

`p(a/c)+q(b/c)+r=0......[3]`

It is obvious from equation [1] and [3] that equation [3] is possible when the variable straight line passes through the fixed point `(a/c,b/c)`as `a,b and c` are constants