Let z=x+iy𝑧=𝑥+𝑖𝑦 and a point P𝑃 represent z𝑧 in the Argand plane. If the real part of z−1z+i𝑧-1𝑧+𝑖 is 1,1, then a point that lies on the locus of P𝑃 is
Question
Let and a point represent in the Argand plane.
If the real part of is 1, then a point that lies on the locus of is...
Solution
The given condition is that the real part of (z - 1)/(z + i) is 1.
Let's write z as x + iy and substitute it into the equation:
Real[(x + iy - 1) / (x + iy + i)] = 1
This simplifies to:
Real[((x - 1) + iy) / (x + (y + 1)i)] = 1
The real part of a complex number a + bi is a. So, we can separate the real and imaginary parts of the complex number:
[(x - 1)x + y(y + 1)] / [x^2 + (y + 1)^2] = 1
Solving this equation for y, we get:
y = -x / (x - 1)
So, any point (x, y) that satisfies this equation lies on the locus of P. For example, the point (2, -2) lies on the locus of P.
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