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Question
Find the value of k, if the points A(7, −2), B (5, 1) and C (3, 2k) are collinear.
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Solution
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(7, −2), B(5, 1) and C(3, 2k). It is also said that they are collinear and hence the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( 7 \times 1 + 5 \times 2k + 3 \times - 2 \right) - \left( 5 \times - 2 + 3 \times 1 + 7 \times 2k \right) \right|\]
\[ 0 = \frac{1}{2}\left| \left( 7 + 10k - 6 \right) - \left( - 10 + 3 + 14k \right) \right|\]
\[ 0 = \frac{1}{2}\left| - 4k + 8 \right|\]
\[ 0 = - 4k + 8\]
\[ k = 2\]
Hence the value of ‘k’ for which the given points are collinear is k = 2 .
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