Not your average complex problem
It is given that for some complex numbers z , ∣ ∣ z − 5 3 − 5 i ∣ ∣ = 5 , which is equivalent to ∣ ∣ ∣ ∣ z 1 − a b 1 + c i ∣ ∣ ∣ ∣ = d 1 , where a , b , c and d are positive integers and b is square-free.
Find a + b + c + d .
Clarification : i = − 1 .
Hint : Consider what happens to z in the complex plane when its reciprocal is taken.
The answer is 38.
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1 solution
Nice question!
Can you explain more with image
The deal is, if you are given a point z in the complex plane, you need to draw the line from the origin to the point, and find on that line such that the product of the distance from the origin to z and the distance of the origin to that point is equal to 1. Then you need to reflect that point over the x-axis.
See what happens to the point?
In the same manner, we also do this to the given circle, as follows:
How did you find out the center and radius of the inverted circle? I did it by solving some pretty complicated equations. Just wondering if there is a simpler way to do it, like a direct formula relate to the center and radius of original circle.
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You invert the two points on the circle on which the line from the origin to the center of the circle passes through. The midpoint of the inverted points becomes the new center and the distance between the inverted points becomes the diameter.
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Got it, I got a simple formula. For the inverted circle, both the coordinates of the center(from the origin) and the radius get scaled by a factor of 1/(a^2-R^2), where a and R are the coordinate and radius of the original circle.
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@Wei Chen – could u please elaborate as to how u arrived at this result
z − 1 in the complex plane is simply the inversion of z over the origin with radius 1 reflected over the x-axis. In our particular case, we are given the equation of a circle in the complex plane centered at ( 5 3 , 5 ) and radius 5 and we want to invert this over the origin with radius 1 and reflect it over the x-axis. The inverted-and-reflected circle then becomes centered at ( 5 3 1 , − 1 5 1 ) and has radius 1 5 1 . Therefore,
∣ z 1 − 5 3 1 + 1 5 i ∣ = 1 5 1
And this gives our answer to be 1 5 + 1 5 + 5 + 3 = 3 8 .
For your reference, here is an article about inversion.