Originally Posted by

**Sean12345** The question is;

Find the equations of the lines which pass through the point $\displaystyle (1,2)$ and have perpendicular distance of $\displaystyle 1$ from the origin.

The question says "lines" and "equations". The plural makes me think I have gone wrong since I only have one. Here is working....

Let $\displaystyle l$ be a line which passes through $\displaystyle (1,2)$ then $\displaystyle l:~y=(2-a)+ax$ for some $\displaystyle a$. Let the line perpendicular to $\displaystyle l$ passing through the origin be $\displaystyle p$ and thus $\displaystyle p:~y=-\frac{1}{a}\cdot x$ .

Let $\displaystyle l$ and $\displaystyle p$ intersect at $\displaystyle R$ then;

$\displaystyle (2-a)+ax=-\frac{1}{a}\cdot x$

$\displaystyle \implies x=\frac{a(a-2)}{a^2+1}~~,~~y=-\frac{(a-2)}{a^2+1}$

Thus $\displaystyle R:~\left(\frac{a(a-2)}{a^2+1}~,~-\frac{(a-2)}{a^2+1}\right)$

Let the distance between $\displaystyle R$ and the origin be $\displaystyle d$ then,

$\displaystyle d=\sqrt{\left(\frac{a(a-2)}{a^2+1}\right)^2+\left(\frac{(a-2)}{a^2+1}\right)^2}=\frac{(a-2)}{\sqrt{a^2+1}}=1$

$\displaystyle \implies (a-2)=\sqrt{a^2+1} \implies a=\frac{3}{2}$