Question

If the roots of the equation $(a^2+b^2)x^2-2(ac+bd)x+c^2+d^2=0$, where a, b, c and $d\ne 0$, are equal, prove that $\frac{a}{b}=\frac{c}{d}$.

Answer 1

$\left(a+b\right)x-2\left(ab+cd\right)x+\left(c+d\right)=0$

equal roots then

$D=b-4ac=0$

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

$4\left(ab+cd\right)-4\left(a+b\right)\left(c+d\right)=0$

$\left(ab+cd+2abcd\right)-ac-ad-bd-bc=0$

$-ac-bd+2abcd=0$

$-\left(ac+bd-2abcd\right)=0$

$\left\{\left(ac-bd\right)\right\}=0$

$ac-bd=0$

$ac=bd$

$\frac{a}{b}=\frac{d}{c}$.