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Pair of Straight Lines — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsPair of Straight Lines3 Marks
Question
Without expanding determinant, show that

$\left|\begin{array}{lll}1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|=10\left|\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 7 \\ 3 & 2 & 6\end{array}\right|$

✓

Answer

L.H.S. $=\left|\begin{array}{ccc}1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|$

In 1st determinant, taking 2 common from $C_3$ we get

L.H.S. $=\left|\begin{array}{ccc}1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|$

In $1^{\text {st }}$ determinant, taking 2 common from $\mathrm{C}_3$,

we get

L.H.S. $=2\left|\begin{array}{lll}1 & 3 & 3 \\ 6 & 1 & 2 \\ 3 & 7 & 6\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|$

$=\left|\begin{array}{lll}2 & 3 & 3 \\ 12 & 1 & 2 \\ 6 & 7 & 6\end{array}\right|+\left|\begin{array}{lll}8 & 3 & 3 \\ 8 & 1 & 2 \\ 4 & 7 & 6\end{array}\right|$

$=\left|\begin{array}{ccc}2+8 & 3 & 3 \\ 12+8 & 1 & 2 \\ 6+4 & 7 & 6\end{array}\right|$

$=\left|\begin{array}{lll}10 & 3 & 3 \\ 20 & 1 & 2 \\ 10 & 7 & 6\end{array}\right|$

Interchanging rows and columns, we get

L.H.S. $=\left|\begin{array}{ccc}10 & 20 & 10 \\ 3 & 1 & 7 \\ 3 & 2 & 6\end{array}\right|$

Taking 10 common from $R_1$, we get

L. H.S $=10\left|\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 7 \\ 3 & 2 & 6\end{array}\right|=$ R.H.S.

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