If - - - - are in GP where - - are roots of the equation ax2 - 2bx - c - 0 and - - are roots of the equation px2 - 2qx - r - 0- then

The correct option is A ac/b^2 = pr/q^2 α, β, γ, δ are in GP where α, β are roots of the equation ax^2 + 2bx + c = 0 and γ, δ are r ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Sum of Infinite Terms

If α, β, γ,...

Question

If $α, β, γ, δ$ are in GP where $α, β$ are roots of the equation $a x^{2} + 2 b x + c = 0$ and $γ, δ$ are roots of the equation $p x^{2} + 2 q x + r = 0,$ then

\frac{a c}{b^{2}} = \frac{p r}{q^{2}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{a c}{b} = \frac{p r}{q}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{a b}{c^{2}} = \frac{p q}{r^{2}}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

None of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $\frac{a c}{b^{2}} = \frac{p r}{q^{2}}$
$α, β, γ, δ$ are in GP where $α, β$ are roots of the equation $a x^{2} + 2 b x + c = 0$ and $γ, δ$ are roots of the equation $p x^{2} + 2 q x + r = 0$
As, $α, β, γ, δ$ are in GP.
$\frac{α}{β} = \frac{γ}{δ}$
Applying componendo and dividendo gives
$\frac{α + β}{α - β} = \frac{γ + δ}{γ - δ}$
$\Rightarrow \frac{- 2 b}{\sqrt{4 b^{2} - 4 a c}} = \frac{- 2 q}{\sqrt{4 q^{2} - 4 p r}}$
$\Rightarrow \frac{b^{2}}{b^{2} - a c} = \frac{q^{2}}{q^{2} - p r}$
$∴ \frac{a c}{b^{2}} = \frac{p r}{q^{2}}$
Hence, option A.

Suggest Corrections

Similar questions

Q. Let

α, β

be the roots of the equation

a x^{2} + 2 b x + c = 0

and

γ, δ

be the roots of the equation

p x^{2} + 2 q x + r = 0

.
If

α, β, γ, δ

are in G.P., then-

If $α, β$ are the roots of the equation a $x^{2}$ +2bx +c = 0 and $α + δ$ , $β + δ$ are the roots of A $x^{2}$ + 2Bx + C= 0 , then $\frac{b^{2} - a c}{B^{2} - A C}$ =

If α, β are the roots of a $x^{2}$ +2bx+c=0 and a+δ, β+δ are the roots of A $x^{2}$ + 2Bx + C=0 then $\frac{b^{2} - a c}{B^{2} - A c}$ is

$α, β$ are the roots of $a x^{2} + b x + c = 0 a n d γ, δ a r e t h e r o o t s o f p x^{2} + q x + r = 0 a n d D_{1}, D_{2}$ be the respective discriminants of these equations. If $α, β, γ, a n d δ$ are in A.P. then $D_{1} : D_{2} = (w h e r e α, β, γ δ, ϵ R & a, b, c, p, q, r ϵ R)$

Q. Let

α, β

be the roots of

a x^{2} + b x + c = 0

and

γ, δ

be the roots of

p x^{2} + q x + r = 0

. If

α, β, γ, δ

are in H.P., then

Join BYJU'S Learning Program

If α,β,γ,δ are in GP where α,β are roots of the equation ax2+2bx+c=0 and γ,δ are roots of the equation px2+2qx+r=0, then

If $α, β, γ, δ$ are in GP where $α, β$ are roots of the equation $a x^{2} + 2 b x + c = 0$ and $γ, δ$ are roots of the equation $p x^{2} + 2 q x + r = 0,$ then