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Question
Find the value of k if points A(k, 3), B(6, −2) and C(−3, 4) 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(k, 3), B(6, −2) and C(−3, 4). It is also said that they are collinear and hence the area enclosed by them should be 0.
\[∆ = \frac{1}{2}\left| \left( k\left( - 2 \right) + 6 \times 4 + \left( - 3 \right) \times 3 \right) - \left( 6 \times 3 + \left( - 3 \right)\left( - 2 \right) + k \times 4 \right) \right|\]
\[ 0 = \frac{1}{2}\left| \left( - 2k + 24 - 9 \right) - \left( 18 + 6 + 4k \right) \right|\]
\[ 0 = \frac{1}{2}\left| - 2k + 15 - 24 - 4k \right|\]
\[ 0 = \frac{1}{2}\left| - 6k - 9 \right|\]
\[ 0 = - 6k - 9\]
\[ k = - \frac{9}{6} = - \frac{3}{2}\]
Hence the value of ‘k’ for which the given points are collinear is `(k = - 3 /2)`.
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