z_1 and z_2 are the roots of 3z^2 + 3z + b = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the quadratic equation $3z^2 + 3z + b = 0$,the sum and product of the roots $z_1$ and $z_2$ are:$z_1 + z_2 = -\frac{3}{3} = -1$$z_1 z_2 = \f

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Given the quadratic equation $3z^2 + 3z + b = 0$,the sum and product of the roots $z_1$ and $z_2$ are:$z_1 + z_2 = -\frac{3}{3} = -1$$z_1 z_2 = \frac{b}{3}$The condition for three complex numbers $z_1, z_2, z_3$ to form an equilateral triangle is $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.Here,the vertices are $z_1, z_2$,and the origin $z_3 = 0$.Substituting $z_3 = 0$ into the condition:$z_1^2 + z_2^2 + 0^2 = z_1 z_2 + z_2(0) + (0)z_1$$z_1^2 + z_2^2 = z_1 z_2$Adding $z_1 z_2$ to both sides:$(z_1 + z_2)^2 - 2z_1 z_2 = z_1 z_2$$(z_1 + z_2)^2 = 3z_1 z_2$Substituting the values of the sum and product of roots:$(-1)^2 = 3 \left(\frac{b}{3}\right)$$1 = b$Thus,the value of $b$ is $1$.

$z_1$ and $z_2$ are the roots of $3z^2 + 3z + b = 0$. If the origin,$A(z_1)$,and $B(z_2)$ form an equilateral triangle,then the value of $b$ is equal to

Similar Questions

If $z, iz$ and $z+iz$ are the vertices of a triangle and if $|z|=4$,then the area (in sq. units) of that triangle is:

The equation $|z+1-i|=|z-1+i|$ represents a (where $z$ is a complex number)

The number of points $z$ on the Argand plane which satisfy the conditions $\operatorname{Re}\left(\frac{z-2}{z-4i}\right)=0$ and $\operatorname{Im}\left(\frac{z-2}{z-4i}\right)=1$ simultaneously is

What are the polar coordinates of the point $(-\sqrt{3}, 1)$?

If $z = \sqrt{2} - i\sqrt{2}$ is rotated through an angle $45^{\circ}$ in the anti-clockwise direction about the origin,then the coordinates of its new position are

Vedclass Test Series

Exam Paper Generator

Online Exam Module