Question

If $a,{\alpha }_1,{\alpha }_2,{\alpha }_{2n-1},b$ are in $A.P.a,{\beta }_1,{\beta }_2,....{\beta }_{2n-1},b$ are in G.P. and $a,{\gamma }_1,{\gamma }_2.....{\gamma }_{2n-1},b$ are in
H.P., where a,b are positive, then the equation ${\alpha }_n{X}^2-{\beta }_{n}X+{\gamma }_n=0$ has

• A. real and equal roots
• B. real and unequal roots
• C. imaginary roots
• D. No roots

Answer 1

The correct option is C imaginary roots

${\alpha }_n=a+n{d}_{1}b=a+2n{d}_1\frac{b-a}{2}=n{d}_1{\beta }_n=a{r}^{n}b=a{r}^{2n}{\left(\frac{b}{a}\right)}^{1/2n}=r{\gamma }_n=\frac{a}{[1+n{d}_{2}a]}\frac{1}{b}=\frac{1}{a}+2n{d}_2\frac{a-b}{2ab}=n{d}_{2}Discriminant={{\beta }_n}^2-4{\alpha }_n{\gamma }_{n}D=a{\left(\frac{b}{a}\right)}^{n/2n}-4[a+\frac{b-a}{2}]\frac{a}{[1+\frac{a-b}{2b}]}D=\sqrt{ab}-4\frac{(a+b)}{2}\frac{2ab}{(a+b)}D=\sqrt{ab}-4ab<0$

Imaginary roots