Question

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

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(1,2) and B(3,6). If the points A, B and P are collinear, then the area of triangle ABP will be zero.
$\therefore \frac{1}{2}\left|\begin{array}{ccc}1& 1& 1\\ 3& 6& 1\\ x& y& 1\end{array}\right|=0\Rightarrow \frac{1}{2}\left[1\right(6-y)-2(3-x)+1(3y-6x\left)\right]=0$

$\Rightarrow 6-y-6+2x+3y-6x=0\Rightarrow 2y-4x=0\Rightarrow y=2x$
Hence, the equation of the line joining the given points is y=2x.

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.