Question

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

Answer 1

I. Let P (x, y) be any point on the line joining points A (1, 2) and B (3, 6). Then, the points A, B, and P are collinear.
Therefore, the area of triangle ABP will be zero.
\frac{1}{2}$$\begin{vmatrix}1&2&1\\\3&6&1\\\x&y&1\end{vmatrix}=0
\Rightarrow$$\frac{1}{2}[1(6-y)-2(3-x)+1(3y-6x)]
=6-y-6+2x+3y-6x=0
$\Rightarrow$ y=2x

Hence, the equation of the line joining the given points is y = 2x.

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II. Let P (x, y) be any point on the line joining points A (3, 1) and B (9, 3). Then, the points A, B, and P are collinear.
Therefore, the area of triangle ABP will be zero.
\frac{1}{2}$$\begin{vmatrix}3&1&1\\\9&3&1\\\x&y&1\end{vmatrix}
\Rightarrow$$\frac{1}{2}[3(3-y)-1(9-x)+1(9y-3x)]
$\Rightarrow$ 9-3y-9+x+9y-3x=0
$\Rightarrow$ x-3y=0

Hence, the equation of the line joining the given points is x − 3y = 0