Find the condition for the lines joining the origin to the points of intersection of the circle x² + y² = a² and the line lx + my =1 to coincide.

$\text{ The given circle is }{\mathrm{x}}^2+{\mathrm{y}}^2={\mathrm{a}}^2\dots \left(1\right)$
$\text{ and the given line is }\mathrm{lx}+\mathrm{my}=1\dots \left(2\right)$
Let A and B be the point of intersection of given line and circle.
By homogenizing the equation (1) with equation (2), we get combined equation of $\overline{\mathrm{OA}} \mathrm{and} \overline{\mathrm{OB}}$ is

$\begin{array}{l}{\mathrm{x}}^2+{\mathrm{y}}^2-{\mathrm{a}}^2(1{)}^2=0\\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-{\mathrm{a}}^2(\mathrm{lx}+\mathrm{my}{)}^2=0\\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-{\mathrm{a}}^2\left({\mathrm{l}}^2{\mathrm{x}}^2+{\mathrm{m}}^2{\mathrm{y}}^2+2\mathrm{lmxy}\right)=0\\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-{\mathrm{a}}^2{\mathrm{l}}^2{\mathrm{x}}^2-{\mathrm{a}}^2{\mathrm{m}}^2{\mathrm{y}}^2-2{\mathrm{a}}^2\mathrm{lmxy}=0\\ \Rightarrow \left(1-{\mathrm{a}}^2{\mathrm{l}}^2\right){\mathrm{x}}^2-2{\mathrm{a}}^2\mathrm{lmxy}+\left(1-{\mathrm{a}}^2{\mathrm{m}}^2\right){\mathrm{y}}^2=0.\end{array}$
Comparing equation (3) with 
${\mathrm{Ax}}^2+2\mathrm{Hxy}+{\mathrm{By}}^2=0 \mathrm{we} \mathrm{get}$
$$\begin{gathered} \mathrm{A}=1-{\mathrm{a}}^2{\mathrm{l}}^2 \\ \begin{array}{l}and 2\mathrm{H}=-2{\mathrm{a}}^2\mathrm{lm}\\ \Rightarrow \mathrm{H}=-{\mathrm{a}}^2\mathrm{lm}\end{array} \\ \mathrm{B}=1-{\mathrm{a}}^2{\mathrm{m}}^2 \end{gathered}$$
Given that (3) represents two coincident lines
then H² = AB
$\begin{array}{l}\Rightarrow {\left(-{\mathrm{a}}^2\mathrm{lm}\right)}^2=\left(1-{\mathrm{a}}^2{\mathrm{l}}^2\right)\left(1-{\mathrm{a}}^2{\mathrm{m}}^2\right)\\ \Rightarrow {\mathrm{a}}^4{\mathrm{l}}^2{\mathrm{m}}^2=1-{\mathrm{a}}^2{\mathrm{l}}^2-{\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{a}}^4{\mathrm{l}}^2{\mathrm{m}}^2\\ \Rightarrow {\mathrm{a}}^2{\mathrm{l}}^2+{\mathrm{a}}^2{\mathrm{m}}^2=1\\ \Rightarrow {\mathrm{a}}^2\left({\mathrm{l}}^2+{\mathrm{m}}^2\right)=1\end{array}$
is the required condition