Convert the following in the polar form: 
i) (1 + 7i) / (2 - i)²
ii) (1 + 3i) / (1 - 2i)

Solution:

We will convert each of the numbers of the form a + bi by rationalizing the denominator and then we convert them into polar form.

i) (1 + 7i) / (2 - i)² = (1 + 7i) / (2 + 1 - 4i)

= (1 + 7i) / (3 - 4i) ·(3 + 4i) / (3 + 4i)

= (3 + 4i + 21i - 28) / (9 + 16)

= (-25 + 25i) / 25

= -1 + i

Let us assume that -1 + i = r (cosθ + sinθ) (Polar form) 

Let r cosθ = - 1 and r sinθ = 1

On squaring and adding, we obtain

r² cos² θ + r² sin² θ = (- 1)² + 1²

⇒ r² (cos² θ + sin² θ) = 1 + 1

⇒ r² = 2

⇒ r = √2 [∵ Conventionally, r > 0]

Therefore,

√2 cosθ = - 1 and √2 sinθ = 1

⇒ cosθ = - 1/√2 and sinθ = 1/√2

Since, θ lies in the quadrant II, θ = π - π/4 = 3π/4

Hence,

- 1 + i = r cosθ + ir sinθ

= √2 cos 3π/4 + i √2 sin 3π/4

Thus, the required polar form is √2(cos 3π/4 + i sin3π/4 )

ii) (1 + 3i) / (1 - 2i) = (1 + 3i) / (1 - 2i) · (1 + 2i) / (1 + 2i)

= (1 + 3i + 2i - 6) / (1 + 4)

= (-5 + 5i) / 5

= -1 + i

Just like the previous one, -1 + i = √2(cos 3π/4 + i sin3π/4 )

Thus, the required polar form is √2(cos 3π/4 + i sin3π/4 )

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NCERT Solutions Class 11 Maths Chapter 5 Exercise ME Question 5

Convert the following in the polar form: i) (1 + 7i) / (2 - i)² ii) (1 + 3i) / (1 - 2i)

Summary:

By converting the given complex numbers in polar form, we get

i) √2(cos 3π/4 + i sin3π/4 ) ii) √2(cos 3π/4 + i sin3π/4 )