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Pair of Straight Lines — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsPair of Straight Lines2 Marks
Question
Prove that $\left|\begin{array}{lll}x+y & y+\mathbf{z} & \mathbf{z}+x \\ \mathbf{z}+x & x+y & y+\mathbf{z} \\ y+\mathbf{z} & \mathbf{z}+x & x+y\end{array}\right|=2\left|\begin{array}{lll}x & y & \mathbf{z} \\ \mathbf{z} & x & y \\ y & \mathbf{z} & x\end{array}\right|$
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Answer

L.H.S. $=\left|\begin{array}{lll}x+y & y+z & z+x \\ z+x & x+y & y+z \\ y+z & z+x & x+y\end{array}\right|$

Applying $R_1 \rightarrow R_1+R_2+R_3$, we get

L.H.S.

$=\left|\begin{array}{ccc}2(x+y+z) & 2(x+y+z) & 2(x+y+z) \\ \mathrm{z}+x & x+y & y+z \\ y+z & z+x & x+y\end{array}\right|$

Taking 2 common from $R_1$, we get

L.H.S. $=2\left|\begin{array}{ccc}x+y+z & x+y+z & x+y+z \\ z+x & x+y & y+z \\ y+z & \mathrm{z}+x & x+y\end{array}\right|$

Applying $R_1 \rightarrow R_1-R_3$, we get

L.H.S. $=2\left|\begin{array}{ccc}x & y & \mathrm{z} \\ \mathrm{z}+x & x+y & y+\mathrm{z} \\ y+\mathrm{z} & \mathrm{z}+x & x+y\end{array}\right|$

Applying $R_3 \rightarrow R_3-R_2$, we get

L.H.S. $=2\left|\begin{array}{lll}x & y & z \\ z & x & y \\ y & z & x\end{array}\right|=$ RHS

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