If | $z_{1} | = | z_{2} | = | z_{3} | = | z_{4} | = 1$ and $z_{1} + z_{2} + z_{3} + z_{4} = 0$
then least value of the expression
$E = | z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2}$ is

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B
8

$E = 2 (| z_{1} |^{2} + | z_{2} |^{2} + | z_{3} |^{2} + | z_{4} |^{2})$
$-_{1} (z_{2} + z_{4}) -_{2} (z_{1} + z_{3}) -_{3} (z_{2} + z_{4}) -_{4} (z_{1} + z_{3}) = 2 (| z_{1} |^{2} + | z_{2} |^{2} + | z_{3} |^{2} + | z_{4} |^{2})$
$- (_{1} +_{3}) (z_{2} + z_{4}) - (_{2} +_{4}) (z_{1} + z_{3}) = 8 + | z_{1} + z_{3} |^{2} + | z_{2} + z_{4} |^{2} \geq 8$
$∵ z_{1} + z_{3} = - (z_{2} + z_{4})$

Suggest Corrections

Similar questions

If | $z_{1} | = | z_{2} | = | z_{3} | = | z_{4} | = 1$ and $z_{1} + z_{2} + z_{3} + z_{4} = 0$
then least value of the expression
$E = | z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2}$ is

Q. If

| z_{1} | = | z_{2} | = | z_{3} | = | z_{4} | = 1

and

z_{1} + z_{2} + z_{3} + z_{4} = 0

, then least value of the expression

E = | z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2}

Q. If

z_{1} + z_{2} + z_{3} + z_{4} = 0

, then the expression

| z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2} - 2 (| z_{1} |^{2} + | z_{2} |^{2} + | z_{3} |^{2} + | z_{4} |^{2})

is equal to

0

, is and only if,

Q. If

| z_{1} | = | z_{2} | = | z_{3} | = | z_{4} |

and

z_{1} + z_{2} + z_{3} + z_{4} = 0

then the points

z_{1}, z_{2}, z_{3}, z_{4}

in the Argand plane are the vertices of a

Q. If

| z_{1} | = | z_{2} | = | z_{3} | = | z_{4} |

then the points representing

z_{1}, z_{2}, z_{3}, z_{4}

are

Join BYJU'S Learning Program

If |z1|=|z2|=|z3|=|z4|=1 and z1+z2+z3+z4=0 then least value of the expression E=|z1−z2|2+|z2−z3|2+|z3−z4|2+|z4−z1|2 is

The correct option is B 8 E=2(|z1|2+|z2|2+|z3|2+|z4|2) −¯z1(z2+z4)−¯z2(z1+z3)−¯z3(z2+z4)−¯z4(z1+z3)=2(|z1|2+|z2|2+|z3|2+|z4|2) −(¯z1+¯z3)(z2+z4)−(¯z2+¯z4)(z1+z3)=8+|z1+z3|2+|z2+z4|2≥8 ∵z1+z3=−(z2+z4)

If | $z_{1} | = | z_{2} | = | z_{3} | = | z_{4} | = 1$ and $z_{1} + z_{2} + z_{3} + z_{4} = 0$
then least value of the expression
$E = | z_{1} - z_{2} |^{2} + | z_{2} - z_{3} |^{2} + | z_{3} - z_{4} |^{2} + | z_{4} - z_{1} |^{2}$ is