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 equation of line is
$3x+y-\lambda =0$
Now for x-intercept , $y=0$
$\Rightarrow x=\frac{\lambda }{3}$
So, the coordinates of A are $(\frac{\lambda }{3},0)$
Now for y-intercept , $x=0$
$\Rightarrow y=\lambda$
So, the coordinates of B are $(0,\lambda )$
Now , Let $C(h,k)$ be the foot of perpendicular from $(0,0)$ on the line
$\frac{h-0}{3}=\frac{k-0}{1}=\frac{\lambda }{10}$....(using formula for finding foot of the perpendicular)
$\Rightarrow h=\frac{3\lambda }{10},k=\frac{\lambda }{10}$
So, the coordinates of C are $(\frac{3\lambda }{10},\frac{\lambda }{10})$

So, $BC:CA=9:1$