Question

If the quadratic equation $(b^2+c^2)x^2-2(a+b)cx+(c^2+a^2)=0$ has equal roots. Then find a relation between $a,b,c$.

Answer 1

Given, the quadratic equation $(b^2+c^2)x^2-2(a+b)cx+(c^2+a^2)=0$ has equal roots.

Then we've the discriminant of the given quadratic equation must be $0$.

Then,

$4(a+b{)}^2{c}^2-4(b^2+c^2\left)\right(a^2+c^2)=0$

or, $(a^2+2ab+b^2)c^2-(b^2{c}^2+a^2{b}^2+c^4+a^2{c}^2)=0$

or, $c^4-2ab{c}^2+a^2{b}^2=0$

or, $(c^2-ab{)}^2=0$

or, $c^2=ab$.

So $a,c,b$ are in G.P.