Question
If $$a,\,b,\,c$$ are any three terms of an $$AP$$ then the line $$ax+by+c=0$$
A.
has a fixed direction
B.
always passes through a fixed point
C.
always cuts intercepts on the axes such that their sum is zero
D.
forms a triangle with the axes whose area is constant
Answer :
always passes through a fixed point
Solution :
If $$a,\,b,\,c$$ are the $${p^{th}},{q^{th}}$$ and $${r^{th}}$$ terms of an $$AP$$ whose first term $$ = \lambda $$ and the common difference $$ = \mu $$ then the line is
$$\eqalign{
& \left\{ {\lambda + \left( {p - 1} \right)\mu } \right\}x + \left\{ {\lambda + \left( {q - 1} \right)\mu } \right\}y + \lambda + \left( {r - 1} \right)\mu = 0 \cr
& {\text{or }}\lambda \left\{ {x + y + 1} \right\} + \mu \left\{ {\left( {p - 1} \right)x + \left( {q - 1} \right)y + r - 1} \right\} = 0, \cr
& {\text{which is of the form }}{L_1} + k{L_2} = 0 \cr} $$