Question

Using determinants, find the equation of the line joining the points
(i) (1, 2) and (3, 6)
(ii) (3, 1) and (9, 3)

Answer 1

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

Let the point P be (x, y). So,
Area of triangle ABP = 0

$$\begin{gathered} \Rightarrow ∆=\frac{1}{2}\left|\begin{array}{ccc}1& 2& 1\\ 3& 6& 1\\ x& y& 1\end{array}\right|=0 \\ \Rightarrow 1\left(6-y\right)-2\left(3-x\right)+1\left(3y-6x\right)=0 \\ \Rightarrow 6-y-6+2x+3y-6x=0 \\ \Rightarrow 2y-4x=0 \\ \Rightarrow y=2x \end{gathered}$$

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

Let the point P be (x, y). So,
Area of triangle ABP = 0

$$\begin{gathered} \Rightarrow ∆=\frac{1}{2}\left|\begin{array}{ccc}3& 1& 1\\ 9& 3& 1\\ x& y& 1\end{array}\right|=0 \\ \Rightarrow 3\left(3-y\right)-1\left(9-x\right)+1\left(9y-3x\right)=0 \\ \Rightarrow 9-3y-9+x+9y-3x=0 \\ \Rightarrow -2x+6y=0 \\ \Rightarrow x=3y \end{gathered}$$