Question

Let $\alpha ,\beta$ be the roots of the equation $a{x}^2+2bx+c=0$ and $\gamma ,\delta$ be the roots of the equation $p{x}^2+2qx+r=0$.
If $\alpha ,\beta ,\gamma ,\delta$ are in G.P., then-

• A. $q^{2}ac=b^{2}pr$
• B. $qac=bpr$
• C. $c^{2}pq=r^{2}ab$
• D. $p^{2}ab=a^{2}qr$

Answer 1

The correct option is A $q^{2}ac=b^{2}pr$

Given roots are in G.P

Let the roots be $a,am,a{m}^2,a{m}^3=\alpha ,\beta ,\gamma ,\delta$. Here m is common ratio

$\Rightarrow a(1+m)=\frac{-2b}{a}$

$\Rightarrow a^{2}m=c/a$

$\Rightarrow a{m}^2(1+m)=-2q/p\Rightarrow m^2\frac{-2b}{a}=\frac{-2q}{p}$

$\Rightarrow m^2=\frac{aq}{pb}$

And $a^2{m}^5=r/p$

$\Rightarrow a^2.m\left(m^4\right)=r/p$

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

$\Rightarrow q^{2}ac=b^{2}pr$