Question

Through a given point $O$ a straight line is drawn to cut two given straight lines in $L$ and $M$. Find the locus of a point $N$ on the variable line such that $ON$ is (a) $A.M.$ (b) $H.M.$ (c) $G.M.$ of $OL$ and $OM$.

Answer 1

Any line through $O(o,o)$ is
$\frac{x}{\cos \theta }=\frac{y}{\sin \theta }=\begin{array}{c}{r}_1\\ OL\end{array},\begin{array}{c}{r}_2\\ OM\end{array},\begin{array}{c}{r}_3\\ ON\end{array}$.
Let the given lines be $l_{1}x+m_{1}y=1$
and $l_{2}x+m_{2}y=1$
$L$ lies on $\left(1\right)$
$\therefore l_1{r}_1\cos \theta +m_1{r}_1\sin \theta =1$.
$\therefore r_1=\frac{1}{l_1\cos \theta +m_1\sin \theta }=OL$
You may apply rule 7 (xiii)p.766.
Similarly $r_2=\frac{1}{l_2\cos \theta +m_2\sin \theta }=OM$.
Let $N$ be $(x,y)=(r_3\cos \theta ,r_3\sin \theta )$
(i) If $ON$ is $A.M.$ of $OL$ and $OM$, then ${2r}_3=r_1+r_2$
${2r}_3=\frac{1}{l_1\cos \theta +m_1\sin \theta }+\frac{1}{l_2\cos \theta +m_2\sin \theta }$
$\therefore 2=\frac{1}{l_{1}x+m_{1}y}+\frac{1}{l_{2}x+m_{2}y}$
is the required locus.
(ii) If $ON$ is $H.M.$ of $OL$ and $OM$, then
$\frac{2}{r_3}=\frac{1}{r_1}+\frac{1}{r_2}$
or $\frac{2}{r_3}=(l_1\cos \theta +m_1\sin \theta )+(l_2\cos \theta +m_2\sin \theta )$
$\therefore 2=(l_{1}x+m_{1}y)+(l_{2}x+m_{2}y)$ by $\left(3\right)$
(iii) If $ON$ is $G.M.$ of $OL$ and $OM$, then
$r_3^2=r_1{r}_2$ or $\frac{r_3}{r_1}.\frac{r_3}{r_2}=1$
or $r_3(l_1\cos \theta +m_1\sin \theta ).r_3(l_2\cos \theta +m_2\sin \theta )=1$
or $(l_1\cos \theta +m_1\sin \theta ).(l_2\cos \theta +m_2\sin \theta )=1$ by $\left(3\right)$