Let a - b - c be three complex numbers whose modulus is 1 - If a - b cos- c sin-0- where -0- -2- thenA- b2-c2-0B- b2-c2-0C- a - be i -D- a-b e-i -

A+bcosα +c sinα = 0 ⇒ a = b cosα c sinα ⇒ |a|^2 = |b cosα +c sinα|^2 ⇒ 1 = (b cosα +c sinα)(b̅cosα + c̅sinα) ⇒ b b̅cos^2 α +c c̅sin^2 α nbsp; +12 (b c̅ +b̅ ...

You visited us 4 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Expansions to Remove Indeterminate Form

Let a , b , c...

Question

Let $a, b, c$ be three complex numbers whose modulus is $1$ . If $a + b cos α + c sin α = 0$ , where $α \in (0, \frac{π}{2})$ , then

b^{2} + c^{2} = 0

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

b^{2} - c^{2} = 0

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

a = - b e^{i α}

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

a = - b e^{- i α}

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

Open in App

Solution

The correct options are
A $b^{2} + c^{2} = 0$
C $a = - b e^{i α}$
D $a = - b e^{- i α}$
$a + b cos α + c sin α = 0 \Rightarrow a = - b cos α - c sin α \Rightarrow | a |^{2} = | b cos α + c sin α |^{2} \Rightarrow 1 = (b cos α + c sin α) (¯ b cos α + ¯ c sin α) \Rightarrow b ¯ b {cos}^{2} α + c ¯ c {sin}^{2} α + \frac{1}{2} (b ¯ c + ¯ b c) sin 2 α = 1 \Rightarrow \frac{1}{2} (b ¯ c + ¯ b c) sin 2 α = 0$
As $α \in (0, \frac{π}{2})$
$\Rightarrow (b ¯ c + ¯ b c) = 0 \Rightarrow \frac{b}{¯ b} = - \frac{c}{¯ c} \Rightarrow b^{2} = - c^{2} \Rightarrow b^{2} + c^{2} = 0 \Rightarrow c = \pm i b$

Using this, we get
$a = - (b cos α \pm i b sin α) = - b e^{\pm i α}$

Suggest Corrections

Similar questions

Q. Let

a, b, c

be three complex numbers satisfying the equation

| z | = 1.

a + b cos α + c sin α = 0

, where

α \in (0, \frac{π}{2})

, then which of the following is incorrect?

Q. Let a,b,c be three points lying on the circle

| z | = 1

and suppose

α \in (0, \frac{π}{2})

be such that a + b cos

α + c s i n

α = 0,

then

Q. If sin α and cos α are the roots of the equation

a x^{2} + b x + c = 0,

then

Q. If

a, b, c

are non zero complex number satisfying

a^{2} + b^{2} + c^{2} = 0

and

∣ ∣ ∣ ∣ ∣ \begin{matrix} b^{2} + c^{2} & a b & a c a b & c^{2} + a^{2} & b c a c & b c & a^{2} + b^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = K a^{2} b^{2} c^{2}

, then

K

is equal to

Q. If

a, b, c

are three complex numbers such that

a^{2} + b^{2} + c^{2} = 0

and

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} b^{2} + c^{2} & a b & a c a b & c^{2} + a^{2} & b c a c & b c & a^{2} + b^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = k a^{2} b^{2} c^{2}

then value of

k

Join BYJU'S Learning Program

Let a,b,c be three complex numbers whose modulus is 1. If a+bcosα+csinα=0, where α∈(0,π2), then

Let $a, b, c$ be three complex numbers whose modulus is $1$ . If $a + b cos α + c sin α = 0$ , where $α \in (0, \frac{π}{2})$ , then