Question

Prove that $△=\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|=0$ if a,b,c are in G.P.

Answer 1

Given $△=\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|=0$

$\Rightarrow a(0-{(b\alpha +c)}^2)-b(0-(a\alpha +b)(b\alpha ++c))+a\alpha +b(b^2\alpha +bc-ac\alpha -bc)=0$

$\Rightarrow -a(b^2{\alpha }^2+c^2+2bc\alpha )+b(ab{\alpha }^2+(ac+b^2)\alpha +bc)+a{b}^2{\alpha }^2-a^{2}c{\alpha }^2+b^3\alpha -abc\alpha =0$

$\Rightarrow (abc-a^{2}c){\alpha }^2+(2b^3-2abc)\alpha +b^{2}c-a{c}^2=0$

$\Rightarrow ac(b-a){\alpha }^2+(2b^3-2abc)\alpha +b^{2}c-a{c}^2=0$

By substituting $b^2=ac$ we get

$=0$