Question

The foot of the perpendicular drawn from the origin, on the line, $3x+y=\lambda (\lambda \ne 0)$ is $P$. If the line meets $x$-axis at $A$ and $y$-axis at $B$, then the ratio $BP:PA$ is :

• A. $1:9$
• B. $1:3$
• C. $3:1$
• D. $9:1$

Answer 1

The correct option is D $9:1$

Let the equation of line is
$3x+y=\lambda ...\left(1\right)$


slope of $AB=-3$
$\therefore$ slope of $OP=\frac{1}{3}$
Equation of $OP$ is
$y-0=\frac{1}{3}(x-0)$
$\Rightarrow x=3y$
Put in equation (1)
$\Rightarrow 9y+y=\lambda$
$\Rightarrow y=\frac{\lambda }{10}$
Suppose $P$ divides $AB$ into $m:n$
$\therefore \frac{3\lambda }{10}=\frac{m\cdot 0+n\cdot \frac{\lambda }{3}}{m+n}$
$\Rightarrow \frac{m}{n}=\frac{1}{9}$
Hence, $BP:AP=n:m=9:1$