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प्रश्न
Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.
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उत्तर
The formula for the area ‘A’ encompassed by three points( x1 , y1) , (x2 , y2) and (x3 , y3) is given by the formula,
\[∆ = \frac{1}{2}\left| \left( x_1 y_2 + x_2 y_3 + x_3 y_1 \right) - \left( x_2 y_1 + x_3 y_2 + x_1 y_3 \right) \right|\]
If three points are collinear the area encompassed by them is equal to 0.
The three given points are A(8,1), B(3,−4) and C(2,k). It is also said that they are collinear and hence the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( 8 \times - 4 + 3 \times k + 2 \times 1 \right) - \left( 3 \times 1 + 2 \times - 4 + 8 \times k \right) \right|\]
\[ 0 = \frac{1}{2}\left| \left( - 32 + 3k + 2 \right) - \left( 3 - 8 + 8k \right) \right|\]
\[ 0 = \frac{1}{2}\left| - 25 - 5k \right|\]
\[k = - 5\]
Hence the value of ‘k’ for which the given points are collinear is k = - 5.
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