Question

If A(x, 2), B(−3, −4) and C(7, −5) are collinear, then the value of x is

(a) −63 (b) 63 (c) 60 (b) −60 [CBSE 2014]

Answer 1

The given points A(x, 2), B(−3, −4) and C(7, −5) are collinear.

$$\begin{gathered} \therefore \mathrm{ar}\left(∆\mathrm{ABC}\right)=0 \\ \Rightarrow \frac{1}{2}\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|=0 \\ \Rightarrow x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow x\left[-4-\left(-5\right)\right]+\left(-3\right)\left(-5-2\right)+7\left[2-\left(-4\right)\right]=0 \\ \Rightarrow x+21+42=0 \\ \Rightarrow x+63=0 \\ \Rightarrow x=-63 \end{gathered}$$

Thus, the value of x is −63.

Hence, the correct answer is option A.