Question

If a line $l$ passes through $(k,2k),(3k,3k)$ and $(3,1)$, $k\ne 0$, then the distance from the origin to the line $l$ is

• A. $\frac{1}{\sqrt{5}}$
• B. $\frac{4}{\sqrt{5}}$
• C. $\frac{3}{\sqrt{5}}$
• D. $\frac{2}{\sqrt{5}}$

Answer 1

Answer: (A)

$\frac{1}{\sqrt{5}}$

A line passes through $(k,2k),(3k,3k)$ and $(3,1)$

$\Rightarrow \frac{3k-2k}{3k-k}=\frac{3k-1}{3k-3}$

$\Rightarrow \frac{k}{2k}=\frac{3k-1}{3k-3}$

i.e. $3k-3=2(3k-1)$

i.e. $3k-3=6k-2$

i.e. $3k=-1$ or $k=\frac{-1}{3}$

Slope of line is $\frac{1}{2}$

The equation of the line becomes $y=\frac{x}{2}+(1-\frac{3}{2})$

i.e. $2y=x-1$

The distance of this line from origin is given by $\frac{1}{\sqrt{1+4}}=\frac{1}{\sqrt{5}}$