Question

If 3 points $A(1,a,b),B(a,2,b),C(a,b,3)$ are collinear, then find $a+b=$?

Answer 1

Since, $A(1,a,b),B(a,2,b),C(a,b,3)$ is co-linear then,

$\left|\begin{array}{ccc}1& a& b\\ a& 2& b\\ a& b& 3\end{array}\right|=0$

$\Rightarrow$ $1(6-b^2)-a(3a-ab)+b(ab-2ab)=0$

$\Rightarrow$ $6-b^2-3a^2+a^{2}b+a{b}^2-2a{b}^2=0$

$\Rightarrow$ $a^{2}b+a{b}^2=-6+b^2+3a^2+2a{b}^2$

$\Rightarrow$ $ab(a+b)=3a^2-6+b^2+2a{b}^2$

$\Rightarrow$ $a+b=\frac{3a^2-6+b^2+2ab}{ab}$