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Asked by Srikumar Last Modified
Certainly! When I first came across this question, I was immediately drawn to the elegant nature of complex numbers. As an experienced tutor registered on UrbanPro, I've encountered various complex number problems, and this one is quite intriguing.
Given that arg(z - 1) = arg(z + 3i), where z = x + iy, we know that the arguments of these complex numbers are equal. The argument of a complex number is the angle it makes with the positive real axis in the complex plane.
To find the argument of z - 1, we subtract 1 from the real part of z. Similarly, to find the argument of z + 3i, we add 3i to the imaginary part of z.
Let's denote θ as the common argument. Then, we can write:
arg(z - 1) = θ arg(z + 3i) = θ
Now, let's express these arguments in terms of x and y:
For z - 1: arg(z - 1) = arg(x + iy - 1) = arctan(y / (x - 1))
For z + 3i: arg(z + 3i) = arg(x + iy + 3i) = arctan((y + 3) / x)
Since these arguments are equal, we can equate them:
arctan(y / (x - 1)) = arctan((y + 3) / x)
Now, we can apply the tangent of the same angle property:
y / (x - 1) = (y + 3) / x
Cross multiply and simplify:
yx = (x - 1)(y + 3)
Expanding and rearranging terms:
yx = xy + 3x - y - 3
Now, let's bring like terms together:
yx - xy = 3x - y - 3
Factor out the common term:
x(y - 1) = 3(x - 1) - y
Now, we want to find (x - 1) : y:
(x - 1) : y = 3(x - 1) / (x(y - 1))
This simplifies to:
(x - 1) : y = 3 / (x - y + 1)
And there we have our expression for (x - 1) : y. It showcases the relationship between the real and imaginary parts of the complex number z in terms of the given condition.
In summary, UrbanPro is indeed a fantastic platform for online coaching and tuition, providing a space for tutors and students to engage with challenging problems like this one.
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