(i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

(i) Equation of the line joining (${x_1},{y_1})$ and $({x_2},{y_2})$

is $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&y&1\\{{x_1}}&{{y_1}}&1\\{{x_2}}&{{y_2}}&1\end{array}} \right| = 0$

$\Rightarrow$ $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&y&1\\1&2&1\\3&6&1\end{array}} \right| = 0$

$\Rightarrow$ $x(2 - 6) - y(1 - 3) + 1(6 - 6) = 0$

$\Rightarrow$ $- 4x + 2y = 0 \Rightarrow 2x - y = 0$

Hence, $y = 2x$ is the required line.

(ii) Equation of the line joining $({x_1},{y_1})$ and $({x_2},{y_2})$
is$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&y&1\\{{x_1}}&{{y_1}}&1\\{{x_2}}&{{y_2}}&1\end{array}} \right| = 0$

$\Rightarrow$ $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&y&1\\3&1&1\\9&3&1\end{array}} \right| = 0$

$\Rightarrow$ $x(1 - 3) - y(3 - 9) + 1(9 - 9) = 0 \Rightarrow - 2x + 6y = 0$

$\Rightarrow$ $- x + 3y = 0 \Rightarrow x - 3y = 0$

Hence, $x - 3y = 0$ is the required line.