If \(Z\) is complex number. Then the locus of the point \(Z\) satisfying arg \([(z-i) /(z+i)]=(\pi / 4)\) is a (a) Circle with center \((-1,0)\) and radius \(\sqrt{2}\) (b) Circle with center \((0,0)\) and radius \(\sqrt{2}\) (c) Circle with center \((0,1)\) and radius \(\sqrt{2}\) (d) Circle with center \((1,1)\) and radius \(\sqrt{2}\)

Rewrite the complex number equation \(arg \left[ \frac{(z-i)}{(z+i)} \right] = \frac{\pi}{4}\) using polar form. Recall that for any complex number \(z\), in polar form \(z = re^{i\theta}\) where \(r\) is the magnitude of \(z\) and \(\theta\) is its argument. We have the equation \(\frac{z-i}{z+i}\), which can be represented as \(\frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}}\), where \(z-i\) is represented by \(r_1 e^{i\theta_1}\) and \(z+i\) is represented by \(r_2 e^{i\theta_2}\). Taking the argument of both sides, we have: \(\theta_1 - \theta_2 = \frac{\pi}{4}\)

Now, let \(z = x + yi\), where \(x\) and \(y\) are real numbers. Substituting this value back into our equation, we have: \(arg \left[ \frac{(x + yi -i)}{(x + yi + i)} \right] = \frac{\pi}{4}\) Simplify the equation to get: \(arg \left[ \frac{(x + (y-1)i)}{(x + (y+1)i)} \right] = \frac{\pi}{4}\)

We know that \(tan(\theta_1 - \theta_2) = tan(\frac{\pi}{4}) = 1\). In the equation, let \(\theta_1\) be the argument of the numerator and \(\theta_2\) be the argument of the denominator. Then we have: \(tan(\theta_1 - \theta_2) = \frac{tan(\theta_1) - tan(\theta_2)}{1 + tan(\theta_1)tan(\theta_2)} = 1\) Now, recall that the tangent of the argument for a complex number is equal to the imaginary part divided by the real part: \(tan(\theta_1) = \frac{y - 1}{x}\) and \(tan(\theta_2) = \frac{y + 1}{x}\) Substitute these values back into the equation: \(\frac{\frac{y - 1}{x} - \frac{y + 1}{x}}{1 + \frac{y - 1}{x} \cdot \frac{y + 1}{x}} = 1\)

By simplifying the equation, we have: \(\frac{-2}{1 + ((y - 1)(y + 1))/x^2} = 1\) Let's cross multiply and further simplify: \(-2x^2 = x^2(y^2 -1)\) Dividing both sides by \(x^2\): \(-2 = y^2 - 1\) Solving for \(y^2\): \(y^2 = 3\)

We now see that the locus of the point \(Z\) is in the form of a circle with center at \((0,0)\) and radius of \(\sqrt{3}\), which is not one of the given options. However, we made an error in Step 4 when dividing by \(x^2\). Let's go back and correct it. Divide both sides by \(-1\): \(2x^2 = -x^2(y^2 - 1)\) Now divide both sides by \(-x^2\) and simplify: \(2 = y^2 - 1\) Solving for \(y^2\): \(y^2 = 1 + 2\) \(y^2 = 3\) Now we can see the correct answer: The locus of the point \(Z\) is in the form of a circle with center at \((0,0)\) and radius of \(\sqrt{3}\), which is still not one of the given options. However, we can eliminate the other choices by the process of elimination: (a) Circle with center \((-1,0)\) and radius \(\sqrt{2}\): This locus is incorrect because the center is not at the origin and the radius is not \(\sqrt{3}\). (b) Circle with center \((0,0)\) and radius \(\sqrt{2}\): This locus is incorrect because the radius is not \(\sqrt{3}\). (c) Circle with center \((0,1)\) and radius \(\sqrt{2}\): This locus is incorrect because the center is not at the origin and the radius is not \(\sqrt{3}\). (d) Circle with center \((1,1)\) and radius \(\sqrt{2}\): This locus is incorrect because the center is not at the origin and the radius is not \(\sqrt{3}\). In conclusion, none of the given options are correct. The locus of the point \(Z\) satisfying the equation is a circle with center \((0,0)\) and radius \(\sqrt{3}\).