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Question
If A(x, 2), B(−3, −4) and C(7, −5) are collinear, then the value of x is
Options
−63
63
60
−60
Solution
The given points A(x, 2), B(−3, −4) and C(7, −5) are collinear.
\[\therefore ar\left( ∆ ABC \right) = 0\]
\[ \Rightarrow \frac{1}{2}\left| x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right| = 0\]
\[ \Rightarrow x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) = 0\]
\[\Rightarrow x\left[ - 4 - \left( - 5 \right) \right] + \left( - 3 \right)\left( - 5 - 2 \right) + 7\left[ 2 - \left( - 4 \right) \right] = 0\]
\[ \Rightarrow x + 21 + 42 = 0\]
\[ \Rightarrow x + 63 = 0\]
\[ \Rightarrow x = - 63\]
Thus, the value of x is −63.
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