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प्रश्न
If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.
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उत्तर
The given points A(−2, 1), B(a, b) and C(4, −1) are collinear.
\[\therefore \text{ ar } \left( ∆ ABC \right) = 0\]
\[ \Rightarrow \frac{1}{2}\left| x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right| = 0\]
\[ \Rightarrow x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) = 0\]
\[\Rightarrow - 2\left[ b - \left( - 1 \right) \right] + a\left( - 1 - 1 \right) + 4\left( 1 - b \right) = 0\]
\[ \Rightarrow - 2b - 2 - 2a + 4 - 4b = 0\]
\[ \Rightarrow - 2a - 6b = - 2\]
\[ \Rightarrow a + 3b = 1 . . . . . \left( 1 \right)\]
Also, it is given that
a − b = 1 .....(2)
Solving (1) and (2), we get
\[b + 1 + 3b = 1\]
\[ \Rightarrow 4b = 0\]
\[ \Rightarrow b = 0\]
Putting b = 0 in (1), we get
\[a + 3 \times 0 = 1\]
\[ \Rightarrow a = 1\]
Hence, the respective values of a and b are 1 and 0.
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