Question

If the points A (-2,1),8 (a,b) and.C (4,-1) are collinear and a -b = 1,find the values of a and b.

Answer 1

If 3 points A, B and C are collinear then the are of $\Delta ABC =0$.

The area of the triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is

$\frac{1}{2}\left|\begin{array}{ccc}1&x_1&y_1\\1&x_2&y_2\\1&x_3&y_3\end{array}\right|$

That is $\frac{1}{2}\left|\begin{array}{ccc}1&-2&1\\1&a&b\\1&4&-1\end{array}\right|=0$.

Or

$\left|\begin{array}{ccc}1&-2&1\\1&a&b\\1&4&-1\end{array}\right|=0 \\ \left|\begin{array}{ccc}0&-6&-2\\0&a-4&b+1\\1&4&-1\end{array}\right|=0 \\ -6(b+1)+2(a-4)=0\\ -3(b+1)+a-4=0\\ a-3b=7 ............................(1)$

Given the equation $a-b=1........................(2)$

(2)-(1) gives

$2b=-6\\ b=-3\\ a=-2$