Question
The foot of the perpendicular on the line $$3x + y = \lambda $$ drawn from the origin is $$C.$$ If the line cuts the $$x$$-axis and $$y$$-axis at $$A$$ and $$B$$ respectively then $$BC : CA$$ is :
A.
1 : 3
B.
3 : 1
C.
1 : 9
D.
9 : 1
Answer :
9 : 1
Solution :
$$\eqalign{ & \tan \left( {{{180}^ \circ } - \theta } \right) = {\text{slope of }}AB = - 3 \cr & \therefore \tan \,\theta = 3 \cr & \therefore \frac{{OC}}{{AC}} = \tan \,\theta ,\,\frac{{OC}}{{BC}} = \cot \,\theta \cr & \Rightarrow \frac{{BC}}{{AC}} = \frac{{\tan \,\theta }}{{\cot \,\theta }} = {\tan ^2}\theta = 9 \cr} $$
$$\eqalign{ & \tan \left( {{{180}^ \circ } - \theta } \right) = {\text{slope of }}AB = - 3 \cr & \therefore \tan \,\theta = 3 \cr & \therefore \frac{{OC}}{{AC}} = \tan \,\theta ,\,\frac{{OC}}{{BC}} = \cot \,\theta \cr & \Rightarrow \frac{{BC}}{{AC}} = \frac{{\tan \,\theta }}{{\cot \,\theta }} = {\tan ^2}\theta = 9 \cr} $$