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Question: If A(z<sub>1</sub>), B(z<sub>2</sub>) and C(z<sub>3</sub>) be the vertices of a triangle ABC in whic...

If A(z1), B(z2) and C(z3) be the vertices of a triangle ABC in which ∠ABC =π4\frac{\pi}{4}and ABBC=2\frac{AB}{BC} = \sqrt{2}, then the value of z2 is equal to

A

z3+ i(z1 + z3)

B

z3 – i(z1 – z3)

C

z3+ i(z1 – z3)

D

None of these

Answer

z3+ i(z1 – z3)

Explanation

Solution

Sol. ABBC=2\frac{AB}{BC} = \sqrt{2}.

Considering the rotation about ‘B’, we get,

z1−z2z3−z2=∣z1−z2∣∣z3−z2∣eiπ/4=ABBCeiπ/4\frac{z_{1} - z_{2}}{z_{3} - z_{2}} = \frac{\left| z_{1} - z_{2} \right|}{\left| z_{3} - z_{2} \right|}e^{i\pi/4} = \frac{AB}{BC}e^{i\pi/4}

= 2(12+i2)=1+i\sqrt{2}\left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right) = 1 + i

⇒ z1 – z2 = (1 + i) (z3 – z2)

⇒ z1 – (1 + i)z3 = z2(1 – 1 – i) = – iz2 ⇒ z2 = iz1 – i (1 + i) z3 = z3 + i (z1 – z3)