Question

If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p² + q²) is equal to ______.

Options

• 7

• 8

• 9

• 10

Answer 1

If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p² + q²) is equal to 10.

Explanation:

Given: z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution.

Put z = z + iy

(x + iy) + α|x + iy – 1| + 2i = 0

∵ Modulus of z = x + iy is |z| = `sqrt(x^2 + "y"^2)`

∴ `(x + "iy") + αsqrt((x - 1)^2 + "y"^2) + 2"i"` = 0

Compare real and imaginary parts,

∴ y + 2 = 0 and `x + αsqrt((x - 1)^2 + "y"^2)` = 0

⇒ y = –2 and `x + αsqrt((x - 1)^2 + 4)` = 0

⇒ x² = α²(x – 1)² + 4α²

⇒ α² = `x^2/(x^2 - 2x + 5)`

⇒ x²(α² – 1) – 2xα² + 5α² = 0 is a quadratic in x whose roots are real.

∴ D ≥ 0

⇒ (2α²)² – 4(α² – 1)(5α²) ≥ 0

⇒ 4α⁴ – 4(α² – 1)(5α²) ≥ 0

⇒ α²(–4α² + 5) ≥ 0

⇒ `α^2(α^2 - 5/4) ≤ 0`

`α^2 ∈ [0, 5/4]`

`α∈[- sqrt(5)/2, sqrt(5)/2]`

p = `-sqrt(5)/2`, q = `sqrt(5)/2`

4(p² + q²) = `4(5/4 + 5/4)` = 10