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If $(a, b),(c, d)$ and $(a-c, b-d)$ are collinear, then which one of the following is correct?
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Verified Answer
The correct answer is:
$\mathrm{bc}-\mathrm{ad}=0$
Let $A, B$ and $C$ having co-ordinates $(a, b),(c, d)$ and $\{(a-c),(b-d)\}$ respectively be the points If these poins are collinear then
$\left|\begin{array}{ccc}a & b & 1 \\ c & d & 1 \\ a-c & b-d & 1\end{array}\right|=0$
$\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ gives
$\left|\begin{array}{ccc}a & b & 1 \\ c-a & d-b & 0 \\ a-c & b-d & 1\end{array}\right|=0$
$\mathrm{R}_{3} \rightarrow \mathrm{R}_{2}+\mathrm{R}_{3}$ gives
$\left|\begin{array}{ccc}a & b & 1 \\ c-a & d-b & 0 \\ 0 & 0 & 1\end{array}\right|=0$
$\Rightarrow 1 .\{\mathrm{a}(\mathrm{d}-\mathrm{b})-\mathrm{b}(\mathrm{c}-\mathrm{a})\}=0$
$\Rightarrow \mathrm{ad}-\mathrm{ab}-\mathrm{bc}+\mathrm{ab}=0$
$\Rightarrow b c-a d=0$
$\left|\begin{array}{ccc}a & b & 1 \\ c & d & 1 \\ a-c & b-d & 1\end{array}\right|=0$
$\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ gives
$\left|\begin{array}{ccc}a & b & 1 \\ c-a & d-b & 0 \\ a-c & b-d & 1\end{array}\right|=0$
$\mathrm{R}_{3} \rightarrow \mathrm{R}_{2}+\mathrm{R}_{3}$ gives
$\left|\begin{array}{ccc}a & b & 1 \\ c-a & d-b & 0 \\ 0 & 0 & 1\end{array}\right|=0$
$\Rightarrow 1 .\{\mathrm{a}(\mathrm{d}-\mathrm{b})-\mathrm{b}(\mathrm{c}-\mathrm{a})\}=0$
$\Rightarrow \mathrm{ad}-\mathrm{ab}-\mathrm{bc}+\mathrm{ab}=0$
$\Rightarrow b c-a d=0$
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