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

Q : 4         (ii) Find equation of line joining \small (3,1) and \small (9,3) using determinants.

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We can find the equation of the line by considering any arbitrary point A(x,y) on line.

So, we have three points which are collinear and therefore area surrounded by them will be equal to zero.

\triangle = \frac{1}{2}\begin{vmatrix} 3 &1 &1 \\ 9& 3 & 1\\ x& y &1 \end{vmatrix} = 0

Calculating the determinant:

=\frac{1}{2}\left [ 3\begin{vmatrix} 3 &1 \\ y& 1 \end{vmatrix}-1\begin{vmatrix} 9 &1 \\ x& 1 \end{vmatrix}+1\begin{vmatrix} 9 &3 \\ x &y \end{vmatrix} \right ]

=\frac{1}{2}\left [ 3(3-y)-1(9-x)+1(9y-3x) \right ] = 0

\frac{1}{2}\left [ 9-3y-9+x+9y-3x \right ] = \frac{1}{2}[6y-2x] = 0

Hence we have the line equation: 

3y= x or  x-3y = 0.

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