Question
If $$a,\,c,\,b$$ are in GP then the line $$ax+by+c=0$$
A.
has a fixed direction
B.
always passes through a fixed point
C.
forms a triangle with the axes whose area is constant
D.
always cuts intercepts on the axes such that their sum is zero
Answer :
forms a triangle with the axes whose area is constant
Solution :
$$\eqalign{
& {\text{The area of the triangle}} = \frac{1}{2} \times \left( {x{\text{ - intercept}}} \right) \times \left( {y{\text{ - intercept}}} \right) \cr
& = \frac{1}{2} \times \left( { - \frac{c}{a}} \right) \times \left( { - \frac{c}{b}} \right) \cr
& = \frac{1}{2}.\frac{{{c^2}}}{{ab}} \cr
& = \frac{1}{2}\,\,\,\,\left( {\because \,\,a,\,c,\,b{\text{ are in GP}}} \right). \cr} $$