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 1

Answer: (D)

$9:1$

Given line is $3x+y=\lambda$
Slope of this line is $-3$
$\tan ({180}^{\circ }-\theta )=$ slope of line $=-3$
$\therefore \tan \theta =3$
Now, in $△OCA$
$\frac{OC}{AC}=\tan \theta$ .....(1)
In $△OCB$
$\tan ({90}^0-\theta )=\frac{OC}{BC}$
$\frac{OC}{BC}=\cot \theta$ .....(2)

Dividing (1) by (2), we get
$\Rightarrow \frac{BC}{AC}=\frac{\tan \theta }{\cot \theta }={\tan }^2\theta =9$

Hence, $BC:AC=9:1$