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Question
Find the value of x if the area of ∆ is 35 square cms with vertices (x, 4), (2, −6) and (5, 4).
Solution
\[∆ = \frac{1}{2}\begin{vmatrix}x & 4 & 1 \\ 2 & - 6 & 1 \\ 5 & 4 & 1\end{vmatrix} = \pm 35\]
\[ = \frac{1}{2}\begin{vmatrix}x & 4 & 1 \\ 2 - x & - 10 & 0 \\ 5 & 4 & 1\end{vmatrix} = \pm 35 \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \frac{1}{2}\begin{vmatrix}x & 4 & 1 \\ 2 - x & - 10 & 0 \\ 5 - x & 0 & 0\end{vmatrix} = \pm 35 \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \frac{1}{2}\begin{vmatrix}2 - x & - 10 \\ 5 - x & 0\end{vmatrix} = \pm 35\]
\[ = 0 + 10\left( 5 - x \right) = \pm 70\]
\[ \Rightarrow 50 - 10x = 70\text{ or }50 - 10x = - 70\]
\[ \Rightarrow - 10x = 20\text{ or }- 10x = - 120\]
\[ \Rightarrow x = - 2\text{ or }x = 12\]
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