Question
If $$0 < a < b < c$$ and the roots $$\alpha ,\beta $$ of the equation $$ax^2 + bx + c = 0$$ are imaginary then incorrect statement is
A.
$$\left| \alpha \right| = \left| \beta \right|$$
B.
$$\left| \alpha \right| > 1$$
C.
$$\left| \beta \right| < 1$$
D.
None of these
Answer :
$$\left| \beta \right| < 1$$
Solution :
Since the roots are imaginary
∴ $$D < 0$$ and roots occur as conjugate pair, i.e., $$\beta = \overline \alpha $$
$$\eqalign{
& \therefore \left| \beta \right| = \left| {\overline \alpha } \right| = \left| \alpha \right| \cr
& {\text{Also, let }}\alpha = \frac{{ - b + i\sqrt {4ac - {b^2}} }}{{2a}} \cr
& \therefore \left| \alpha \right| = \sqrt {\frac{{{b^2}}}{{4{a^2}}} + \frac{{4ac - {b^2}}}{{4{a^2}}}} = \sqrt {\frac{c}{a}} \cr
& \left| \alpha \right| > 1\left( {\because c > a} \right) \cr
& \therefore \left| \alpha \right| = \left| \beta \right| > 1 \cr} $$