Let be the distinct solutions of the equation . The is equal to : [2026]

Question

Let $z_{1}, z_{2} \in C$ be the distinct solutions of the equation $z^{2} + 4 z - (1 + 12 i) = 0$ . The $| z_{1} |^{2} + | z_{2} |^{2}$ is equal to : [2026]

Smart Achievers · Accepted Answer

(4)

Given, $z^{2} + 4 z - (1 + 12 i) = 0$

$\Rightarrow z = \frac{- 4 \pm \sqrt{16 + 4 (1 + 12 i)}}{2}$

$\Rightarrow z = \frac{- 4 \pm \sqrt{20 + 48 i}}{2}$

$\Rightarrow z = - 2 \pm \sqrt{5 + 12 i}$

$∴ z_{1} = - 2 + \sqrt{5 + 12 i} and z_{2} = - \sqrt{5 + 12 i}$

But $\sqrt{5 + 12 i} = \pm (3 + 2 i)$

$∴ z_{1} = - 2 + (3 + 2 i) and z_{2} = - 2 - (3 + 2 i)$

$\Rightarrow z_{1} = 1 + 2 i and z_{2} = - 5 - 2 i$

Now, $| z_{1} | = \sqrt{1 + 4} = \sqrt{5} and | z_{2} | = \sqrt{25 + 4} = \sqrt{29}$

$∴ | z_{1} |^{2} + | z_{2} |^{2} = (5) + (29) = 34$

Solution

Q.
Let $z_{1}, z_{2} \in C$ be the distinct solutions of the equation $z^{2} + 4 z - (1 + 12 i) = 0$ . The $| z_{1} |^{2} + | z_{2} |^{2}$ is equal to : [2026]

1 18

2 22

3 29

4 34

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