Question

If the sum of square of roots of the equation $x^2+(p+iq)x+3i=0$ is 8, then find the value of $p$ and $q$, where $p$ and $q$ are real.

• A. $p=3$ and $q=1$
• B. $p=-3$ and $q=1$
• C. $p=-3$ and $q=-1$
• D. $p=3$ and $q=-1$

Answer 1

Answer: (A), (C)

The correct options are
A $p=3$ and $q=1$
C $p=-3$ and $q=-1$

Let the roots be $\alpha$ and $\beta$.
We have $\alpha +\beta =-(p+iq)$; $\alpha \beta =3i$
Given: ${\alpha }^2+{\beta }^2=8$
or $(\alpha +\beta {)}^2-2\alpha \beta =8$
or $(p+iq{)}^2-6i=8$
or $p^2-q^2+i(2pq-6)=8$
or $p^2-q^2=8$ and $pq=3$
or $p=3$ and $q=1$ or $p=-3$ and $q=-1$
Ans: A,C