Let z 1 and z 2 be roots of the equation z 2- pz - q -0- p - q - c-Let A and B represents z 1 and z 2 in the complex plane- If - AoB -0- AND OA - OB - O is the origin- then p 2-4 q -A- sin 2-2B- tan 2-2C- cos 2-2D- None of these

The correct option is C z^2+pz+q = 0 ⇒ z_1+z_2 = p,z_1z_2= q OA = OB ⇒ | z_1 |= | z_2 | z_2/z_1 = e^iα = cos α +isin α ⇒z_1+z_2/z_1 = 1+cos α+isin α = 2cos ...

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Byju's Answer

Standard XII

Mathematics

Definition of Functions

Let z 1 and z...

Question

Let $z_{1}$ and $z_{2}$ be roots of the equation $z^{2} + p z + q$ = 0,p,q $\in$ c,Let A and B represents $z_{1}$ and $z_{2}$ in the complex plane. If $∠ A o B$ = $α$ $\neq$ 0, AND OA = OB;O is the origin, then $\frac{p^{2}}{4 q}$ =

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None of these

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Solution

The correct option is C

$z^{2} + p z + q$ = 0

$\Rightarrow z_{1} + z_{2}$ = -p, $z_{1} z_{2}$ = q

OA = OB $\Rightarrow | z_{1} | = | z_{2} |$

$\frac{z_{2}}{z_{1}}$ = $e^{i α}$ = $c o s α + i s i n α$

$\Rightarrow \frac{z_{1} + z_{2}}{z_{1}}$ = $1 + c o s α + i s i n α$

= $2 c o s \frac{α}{2} (c o s \frac{α}{2} + i s i n \frac{α}{2})$

= $\frac{(z_{1} + z_{2})^{2}}{z_{1}^{2}}$ = $4 c o s^{2} \frac{α}{2} e^{i α}$ = $4 c o s^{2} \frac{α}{2} \frac{z_{2}}{z_{1}}$

= $(z_{1} + z_{2})^{2}$ = $4 c o s^{2} \frac{α}{2} z_{1} z_{2}$

$\Rightarrow p^{2}$ = $4 q c o s^{2} \frac{α}{2}$

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Let z1 and z2 be roots of the equation z2+pz+q = 0,p,q∈c,Let A and B represents z1 and z2 in the complex plane. If ∠AoB = α ≠ 0, AND OA = OB;O is the origin, then p24q =

The correct option is C z2+pz+q = 0 ⇒z1+z2 = -p,z1z2= q OA = OB ⇒|z1|=|z2| z2z1 = eiα = cos α+isin α ⇒z1+z2z1 = 1+cos α+isin α = 2cosα2(cosα2+isinα2) = (z1+z2)2z21 = 4cos2α2eiα = 4cos2α2z2z1 = (z1+z2)2 = 4cos2α2z1z2 ⇒p2 = 4qcos2α2

Let $z_{1}$ and $z_{2}$ be roots of the equation $z^{2} + p z + q$ = 0,p,q $\in$ c,Let A and B represents $z_{1}$ and $z_{2}$ in the complex plane. If $∠ A o B$ = $α$ $\neq$ 0, AND OA = OB;O is the origin, then $\frac{p^{2}}{4 q}$ =