Find the Locus of mid-point of line segment intercepted between real and imaginary Axis, by the line a’ z + a z’ +b =0 where b is real parameter and a is fixed complex number such that Re(a) and Im(z) is not equal to zero . a’ and z’ are complex conjugate of a and z respectively .

Ans:
Hello Student,
Please find solution to your question below

a’z + az + b = 0
Let z = x + iy
z’ = x – iy
Let a = c + id, where c & d are real numbers.
a’ = c – id
Put in the equation, we have
(c – id)(x + iy) + (c + id)(x – iy) + b = 0
(cx + dy + i(cy - dx)) + (cx + dy + i(-cy + dx)) + b = 0
2cx + 2dy + b = 0
Let the mid – point of the line segment to be (h, k)
Line intersect the axis at
Real Axis:
y = 0

Imaginary Axis:
x = 0