Using determinants, find the equation of the line joining the points

(i)

Given: $A=(1,2)$ and $B=(3,6)$

Let the point $P$ be $(x, y) .$ So,

Area of triangle $A B P=0$

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

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

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

$\Rightarrow 2 y-4 x=0$

$\Rightarrow y=2 x$

(ii)

Given: $A=(3,1)$ and $B=(9,3)$

Let the point $P$ be $(x, y) .$ So,

Area of triangle $A B P=0$

$\Rightarrow \Delta=\frac{1}{2}\left|\begin{array}{lll}3 & 1 & 1 \\ 9 & 3 & 1 \\ x & y & 1\end{array}\right|=0$

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

$\Rightarrow 9-3 y-9+x+9 y-3 x=0$

$\Rightarrow-2 x+6 y=0$

$\Rightarrow x=3 y$