Let z 1, z 2 be the roots of the equation z 2+ az + 12=0 and z 1, z 2 form an equilateral triangle with origin. Then, the value of |a| is

Question

Let z 1, z 2 be the roots of the equation z 2+ az + 12=0 and z 1, z 2 form an equilateral triangle with origin. Then, the value of |a| is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

text If 0, z, z2 text are vertices of equilateral triangles ⇒ a2+z12+z22=0(z1+z2)+z1 z2 ⇒(z1+z2)2=3 z1 z2 ⇒ a2=3 × 12 ⇒|a|=6

Q. 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

$If 0, z, z_{2} are vertices of equilateral triangles$
$\Rightarrow a^{2} + z_{1}^{2} + z_{2}^{2} = 0 (z_{1} + z_{2}) + z_{1} z_{2}$
$\Rightarrow (z_{1} + z_{2})^{2} = 3 z_{1} z_{2}$
$\Rightarrow a^{2} = 3 \times 12$
$\Rightarrow ∣ a ∣ = 6$

Q. Let z1​,z2​ be the roots of the equation z2+az+ 12=0 and z1​,z2​ form an equilateral triangle with origin. Then, the value of ∣a∣ is

If 0,z,z2​ are vertices of equilateral triangles ⇒a2+z12​+z22​=0(z1​+z2​)+z1​z2​ ⇒(z1​+z2​)2=3z1​z2​ ⇒a2=3×12 ⇒∣a∣=6

Q. 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

$If 0, z, z_{2} are vertices of equilateral triangles$
$\Rightarrow a^{2} + z_{1}^{2} + z_{2}^{2} = 0 (z_{1} + z_{2}) + z_{1} z_{2}$
$\Rightarrow (z_{1} + z_{2})^{2} = 3 z_{1} z_{2}$
$\Rightarrow a^{2} = 3 \times 12$
$\Rightarrow ∣ a ∣ = 6$