If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then (a/c), (b/a) and (c/b) are in

Question

If the sum of the roots of the quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then (a/c), (b/a) and (c/b) are in (A) arithmetic progression (B) geometric progression (C) harmonic progression (D) arithmetic−geometric−progression

Tardigrade Admin · Accepted Answer

Î± + Î² =(1/α2)+(1/β2) Î± + Î² =(α2+β2-2αβ/(Î± + Î² )) (-(b/a))=(b2-2ac/c2) ⇒ 2a2c=b(a2+bc) ⇒ (a/c), (b/a), (c/b) are in H.P.

Q. If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $\frac{a}{c}, \frac{b}{a}$ and $\frac{c}{b}$ are in

arithmetic progression

geometric progression

harmonic progression

arithmetic−geometric−progression

$α + β = \frac{1}{α ^{2}} + \frac{1}{β ^{2}} < b r / > α + β = \frac{α ^{2} + β ^{2} - 2 α β}{( α + β )}$
$(- \frac{b}{a}) = \frac{b ^{2} - 2 a c}{c ^{2}}$
$\Rightarrow 2 a^{2} c = b (a^{2} + b c)$
$\Rightarrow \frac{a}{c}, \frac{b}{a}, \frac{c}{b}$ are in $H . P .$

Q. If the sum of the roots of the quadratic equation ax2+bx+c=0 is equal to the sum of the squares of their reciprocals, then ca​,ab​ and bc​ are in