Question

If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.      

Answer 1

The given points A(−2, 1), B(a, b) and C(4, −1) are collinear.

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

\[\Rightarrow - 2\left[ b - \left( - 1 \right) \right] + a\left( - 1 - 1 \right) + 4\left( 1 - b \right) = 0\]

\[ \Rightarrow - 2b - 2 - 2a + 4 - 4b = 0\]

\[ \Rightarrow - 2a - 6b = - 2\]

\[ \Rightarrow a + 3b = 1 . . . . . \left( 1 \right)\]

Also, it is given that

a − b = 1               .....(2)

Solving (1) and (2), we get

\[b + 1 + 3b = 1\]
\[ \Rightarrow 4b = 0\]
\[ \Rightarrow b = 0\]

Putting b = 0 in (1), we get

\[a + 3 \times 0 = 1\]

\[ \Rightarrow a = 1\]

Hence, the respective values of a and b are 1 and 0.