Question

If the points A(−1, −4), B(b, c) and C(5, −1) are collinear and 2b + c = 4, find the values of b and c.

Answer 1

The given points A(−1, −4), B(b, c) and C(5, −1) are collinear. 

\[\therefore ar\left( ∆ ABC \right) = 0\]
\[ \Rightarrow \frac{1}{2}\left| x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) \right| = 0\]
\[ \Rightarrow x_1 \left( y_2 - y_3 \right) + x_2 \left( y_3 - y_1 \right) + x_3 \left( y_1 - y_2 \right) = 0\]

\[\Rightarrow - 1\left[ c - \left( - 1 \right) \right] + b\left[ - 1 - \left( - 4 \right) \right] + 5\left( - 4 - c \right) = 0\]
\[ \Rightarrow - c - 1 + 3b - 20 - 5c = 0\]
\[ \Rightarrow 3b - 6c = 21\]
\[ \Rightarrow b - 2c = 7 . . . . . \left( 1 \right)\]

Also, it is given that

2b + c = 4               .....(2)

Solving (1) and (2), we get

\[2\left( 7 + 2c \right) + c = 4\]
\[ \Rightarrow 14 + 4c + c = 4\]
\[ \Rightarrow 5c = - 10\]
\[ \Rightarrow c = - 2\]

Putting c = −2 in (1), we get

\[b - 2 \times \left( - 2 \right) = 7\]
\[ \Rightarrow b = 7 - 4 = 3\]

Hence, the respective values of b and c are 3 and −2.