Complex Numbers
Circle-line intersection in complex plane
nta_pyq_2025_apr
Grade 12
Question:
Let$A = {z$$\i_n C$: |$z - 2 - i$| = 3},$B = {z$$\i_n C$: Re($z - iz) = 2}$and$S = A$$\cap B. Then$$\sum is equal to 2$|z| z$\inS ________. 2$
Step-by-Step Solution
Key Concept: Write$z=x+iy$to convert the circle and line conditions into coordinate equations, then sum the required values.
Let$z = x + iy$$(22)$A: |$z - 2 - i$| = 3 |($x - 2) +$(y - 1)$i$| = 3 2 2 ($x - 2) +$(y - 1)$= 9.$...$(1)$.$B = Re(z - iz) = 2$Re(($x + y) + i(y - x)) = 2$$x + y = 2....(2)$On solving$(1)$and$(2)$we get 3 $\pm$$\sqrt17 1$∓$\sqrt17 x =,$y = 2$2 2 1$$\sum$|z| = [2 $\times$$26 + 2$$\times$ 18] 4 z$\i_n s 88$$\Rightarrow$ = 22 4
Correct Answer: 22