Let the point P represent z=x+iy, x, y∈R inthe Argand plane. Let the curves C1 and C2 be the loci of P satisfying the conditions (i) 2z+iz-2 is purely imaginary and (ii) Argz+iz+1=π2, respectively. Then the point of intersection of the curves C1 and C2, other than the origin, is

AP EAMCET · Maths · Complex Number

Let the point $P$ represent $z = x + i y, x, y \in [R in$ the Argand plane. Let the curves $C_{1}$ and $C_{2}$ be the loci of $P$ satisfying the conditions $(i)$ $\frac{2 z + i}{z - 2}$ is purely imaginary and $(ii)$ $Arg (\frac{z + i}{z + 1}) = \frac{π}{2}$ , respectively. Then the point of intersection of the curves $C_{1}$ and $C_{2},$ other than the origin, is

A $(1, 2)$
B $(\frac{2}{7}, - \frac{5}{7})$
C $(- 3, 4)$
D $(\frac{5}{37}, - \frac{30}{37})$

Verified Solution

Answer & Solution

Correct Answer

(D) $(\frac{5}{37}, - \frac{30}{37})$

Step-by-step Solution

Detailed explanation

It is given that z=x+iy, x, y∈ℝ And 2z+iz-2=2(x+iy)+i(x-2)+iy Re2z+iz-2=2x+i(2y+1)(x-2)+iy×(x-2)-iy(x-2)-iy 0=2 x(x-2)+y(2 y+1)=0 0=x2+y2-2x+y2 ...i And, z+iz-i=x+i(y+1)(x+1)+iy×(x+1)-iy(x+1)-iy Now,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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