Question

If the points P(−3, 9), Q(a, b) and R(4, −5) are collinear and a + b = 1, find the values of a and b. [CBSE 2014]

Answer 1

Let A(x₁ = −3, y₁ = 9), B(x₂ = a, y₂ = b) and C(x₃ = 4, y₃ = −5) be the given points.
The given points are collinear if
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow \left(-3\right)\left(b+5\right)+a\left(-5-9\right)+4\left(9-b\right)=0 \\ \Rightarrow -3b-15-14a+36-4b=0 \\ \Rightarrow 2a+b=3 \end{gathered}$$
Now, solving a + b = 1 and 2a + b = 3, we get a = 2 and b = −1.
Hence, a = 2 and b = −1.