Complex Numbers
Modulus one algebraic equation
nta_pyq_2025_apr
Grade 12
Question:
Let z be a complex number such that |z| = 1. If$2+k$$z = kz$, k$\i_n R, then the maximum distance of$k + ik$from 2$¯$k+ z$the circle |$z - (1 + 2i)$| = 1 is:
$\sqrt{5} + 1$
$2$
$3$
$\sqrt{3} + 1$
Step-by-Step Solution
Key Concept: Use$|z|=1$so that powers or reciprocals of$z$can be simplified via$\overline z=1/z$.
2$2 + k$$z = kz$$(1)$$k + z$¯ ¯¯ 2 |z|$k = 2$$k = 2$point p$(2, 4)$; center$(1, 2)$distance from circle ($x - 1)$$2 + (y - 2)$$2 = 1$is max. if ($OP + r)$=$\$sqrt1 + 4 + 1$=$$\sqrt{5} + 1$
Correct Answer: 1