Solveeit Logo
Login

Question

Question: Z<sub>1</sub>¹ Z<sub>2</sub> are two points in an Argand plane. If a\|Z<sub>1</sub>\| = b\|Z<sub>2</...

Z1¹ Z2 are two points in an Argand plane. If a|Z1| = b|Z2|, then the point aZ1−bZ2aZ1+bZ2\frac{aZ_{1} - bZ_{2}}{aZ_{1} + bZ_{2}}is –

A

In the I quadrant

B

In the III quadrant

C

On the real axis

D

On the imaginary axis

Answer

On the imaginary axis

Explanation

Solution

Sol. If Z1 = r1 eiq, Z2= r2ei(q + a), ar1 = br2 and

aZ1−bZ2aZ1+bZ2\frac{aZ_{1} - bZ_{2}}{aZ_{1} + bZ_{2}}= eiθ−ei(θ+α)eiθ+ei(θ+α)\frac{e^{i\theta} - e^{i(\theta + \alpha)}}{e^{i\theta} + e^{i(\theta + \alpha)}}= 1−eiα1+eiα\frac{1 - e^{i\alpha}}{1 + e^{i\alpha}}

= e−iα/2−eiα/2e−iα/2+eiα/2\frac{e^{- i\alpha/2} - e^{i\alpha/2}}{e^{- i\alpha/2} + e^{i\alpha/2}}

= −2isin⁡α22cos⁡α2\frac{- 2i\sin\frac{\alpha}{2}}{2\cos\frac{\alpha}{2}}= –i tan α2\frac{\alpha}{2}

implying that the point is on the imaginary axis.