Question

Find the equation of the line joining (3,1) and (9,3) using determinants.

Answer 1

Let P(x,y) be any point on the line joining A(3,1) and B(9,3). Then, the points A, B and P are collinear, Therefore, then the area of triangle ABP will be zero.
$\therefore \frac{1}{2}\left|\begin{array}{ccc}3& 1& 1\\ 9& 1& 1\\ x& y& 1\end{array}\right|=0$

$\Rightarrow \frac{1}{2}|3(3-y)-1(9-x)+1(9y-3x)|=0$
$\Rightarrow 9-3y-9+x+9y-3x=0\Rightarrow 6y-2x=0\Rightarrow x-3y=0$
Hence, the equation of the line joining the given points is x-3y=0.