Question

Let S = {z ∈ C: |z – 2| ≤, 1, z(1 + i) + z̄(1 – i) ≤ 2}. Let |z – 4i| attains minimum and maximum values, respectively, at z₁ ∈ S and z₂ ∈ S. If 5(|z₁|² + |z₂|²) = α + β`sqrt(5)`, where α and β are integers, then the value of α + β is equal to ______.

पर्याय

• 23

• 24

• 25

• 26

Answer 1

Let S = {z ∈ C: |z – 2| ≤, 1, z(1 + i) + z̄(1 – i) ≤ 2}. Let |z – 4i| attains minimum and maximum values, respectively, at z₁ ∈ S and z₂ ∈ S. If 5(|z₁|² + |z₂|²) = α + β`sqrt(5)`, where α and β are integers, then the value of α + β is equal to 26.

Explanation:

Given: S = {z ∈ C: |z – 2| ≤, 1,z(1 + i) + z̄(1 – i) ≤ 2}

Let z = x + iy

Now, |z – 2| ≤ 1

⇒ |(x – 2) + iy| ≤ 1

⇒ (x – 2)² + y² ≤ 1

⇒ It represents the region inside the circle whose centre is (2, 0) and radius is 1.

Now, z(1 + i) + z̄(1 – i) ≤ 2

⇒ (x + iy)(1 + i) + (x – iy)(1 – i) ≤ 2

⇒ x – y – 1 ≤ 0

⇒ It represents the all points which lies on and above the line x – y – 1 = 0

Now, |z – 4i| represents distance of a point A(0, 4) from z.

Now, AP = `sqrt(17)` and AQ = `sqrt(13)`

∴ |z – 4i|_max = AP and |z – 4i|_min = AD

Let coordinates of point D be (cosθ + 2, sinθ)

Now, (m)_AC = tanθ = –2

⇒ cosθ = `-1/sqrt(5)` and sinθ = `2/sqrt(5)`

∴ Coordinates of point D is `(2 - 1/sqrt(5), 2/sqrt(5))`

So, z₁ = `2 - 1/sqrt(5) + i2/sqrt(5)` and z₂ = 1

Now, |z₁| = `sqrt((2 - 1/sqrt(5))^2 + (2/sqrt(5))^2`

⇒ |z₁| = `sqrt(4 + 1/5 - 4/sqrt(5) + 4/5)`

⇒ |z₁| = `sqrt((5sqrt(5 - 4))/5`

⇒ |z₁|² = `(5sqrt(5) - 4)/sqrt(5)`

Now, 5(|z₁|² + |z₂|²) = `5((5sqrt(5) - 4)/sqrt(5) + 1)`

= `30 - 4sqrt(5)`

⇒ α + β`sqrt(5) = 30 - 4sqrt(5)`

⇒ α = 30, β = –4

∴ α + β = 26