Question

Find the value of k, if the points A (8, 1) B(3, −4) and C(2, k) are collinear.

 

Answer 1

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.