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. [CBSE 2014]

Answer 1

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

$$\begin{gathered} \therefore \mathrm{ar}\left(∆\mathrm{ABC}\right)=0 \\ \Rightarrow \frac{1}{2}\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|=0 \\ \Rightarrow x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow -1\left[c-\left(-1\right)\right]+b\left[-1-\left(-4\right)\right]+5\left(-4-c\right)=0 \\ \Rightarrow -c-1+3b-20-5c=0 \\ \Rightarrow 3b-6c=21 \\ \Rightarrow b-2c=7.....\left(1\right) \end{gathered}$$

Also, it is given that

2b + c = 4 .....(2)

Solving (1) and (2), we get

$$\begin{gathered} 2\left(7+2c\right)+c=4 \\ \Rightarrow 14+4c+c=4 \\ \Rightarrow 5c=-10 \\ \Rightarrow c=-2 \end{gathered}$$

Putting c = −2 in (1), we get

$$\begin{gathered} b-2\times \left(-2\right)=7 \\ \Rightarrow b=7-4=3 \end{gathered}$$

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