∣z1∣ and ∣z2∣ are two distinct points in an Argand plane. If a∣z1∣=b∣z2∣ (where a,b∈R), then the point (az1/bz2)+(bz2/az1) is a point on the

a∣z1∣=b∣z2∣(wherea,b∈R) ...(1)z1=r1(cosA+isinA )&z2=r2(cosB+isinB ) Since, a∣z1∣=b∣z2∣ ⇒ar1=br2let z1=r1(cosA+isinA )=r1cisA&z2=r2(cosB+isinB )=r2cisBw=(bz2az1 )+(az1bz2 )=(br2cisBar1cisA )+(ar1cisAbr2cisB ) ⇒w=cis(A−B)+cis(−A+B)=2cos(A−B) ...{∵ar1=br2}Since w is real.Therefore w lies on a real axis.Hence, option 'A' is correct.

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$∣ z_{1} ∣$ and $∣ z_{2} ∣$ are two distinct points in an Argand plane. If $a ∣ z_{1} ∣ = b ∣ z_{2} ∣$ (where $a, b \in R$ ), then the point $(a z_{1} / b z_{2}) + (b z_{2} / a z_{1})$ is a point on the

line segment [-2, 2] of the real axis

line segment [-2, 2] of the imaginary axis

unit circle

∣ z ∣

= 1

the line with arg z = tan

^{- 1}

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∣ z_{1} ∣

and

∣ z_{2} ∣

are two distinct points in an Argand plane. If

a ∣ z_{1} ∣ = b ∣ z_{2} ∣

(where

a, b \in R

), then the point

(a z_{1} / b z_{2}) + (b z_{2} / a z_{1})

is a point on the

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Text solutionVerified

a ∣ z_{1} ∣ = b ∣ z_{2} ∣ (w h ere a, b \in R)

...(1)

z_{1} = r_{1} (cos A + i sin A) & z_{2} = r_{2} (cos B + i sin B)

Since,

a ∣ z_{1} ∣ = b ∣ z_{2} ∣

\Rightarrow a r_{1} = b r_{2}

let

z_{1} = r_{1} (cos A + i sin A) = r_{1} c i s A & z_{2} = r_{2} (cos B + i sin B) = r_{2} c i s B

w = (\frac{a z _{1}}{b z _{2}}) + (\frac{b z _{2}}{a z _{1}}) = (\frac{a r _{1} c i s A}{b r _{2} c i s B}) + (\frac{b r _{2} c i s B}{a r _{1} c i s A})

\Rightarrow w = c i s (A - B) + c i s (- A + B) = 2 cos (A - B)

...{ $∵ a r_{1} = b r_{2}$ }

Since

w

is real.
Therefore

w

lies on a real axis.
Hence, option 'A' is correct.

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Question Text	$∣ z_{1} ∣$ and $∣ z_{2} ∣$ are two distinct points in an Argand plane. If $a ∣ z_{1} ∣ = b ∣ z_{2} ∣$ (where $a, b \in R$ ), then the point $(a z_{1} / b z_{2}) + (b z_{2} / a z_{1})$ is a point on the
Answer Type	Text solution:1
Upvotes	150

∣z1​∣ and ∣z2​∣ are two distinct points in an Argand plane. If a∣z1​∣=b∣z2​∣ (where a,b∈R), then the point (az1​/bz2​)+(bz2​/az1​) is a point on the

Text solutionVerified

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$∣ z_{1} ∣$ and $∣ z_{2} ∣$ are two distinct points in an Argand plane. If $a ∣ z_{1} ∣ = b ∣ z_{2} ∣$ (where $a, b \in R$ ), then the point $(a z_{1} / b z_{2}) + (b z_{2} / a z_{1})$ is a point on the