JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Complex Numbers

Let

z_{1}, z_{2}

be the roots of the equation

z^{2} + a z + 12 = 0

and

z_{1}, z_{2}

form an equilateral triangle with origin. Then, The value of

∣ a ∣

is ____

Answer: 6

Solution

Three complex numbers

z_{1}

z_{2}

z_{3}

form an equilateral triangle exactly when

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}

With the third vertex at the origin this reduces to

z_{1}^{2} + z_{2}^{2} = z_{1} z_{2}

. Adding

2 z_{1} z_{2}

to both sides gives

(z_{1} + z_{2})^{2} = 3 z_{1} z_{2}

From the quadratic,

z_{1} + z_{2} = - a

and

z_{1} z_{2} = 12

, so

a^{2} = 36

and

∣ a ∣ = 6

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Integer answer