This discussion focuses on finding the center and radius of a circle defined by the complex equation args((z-3i)/(z+4))=π/6. The correct radius, determined through Pythagorean theorem, is confirmed to be 5 unit². Participants discussed expressing the complex ratio in terms of real and imaginary components, leading to the identification of points on the circle at coordinates (0,3) and (-4,0). The solution emphasizes the importance of transforming complex expressions to derive geometric properties accurately.
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Kajan thana said:Homework Statement
Sketch the loci, find centre point and the radius of the circle.
args((z-3i)/((z+4))=π/6[/B]
Homework Equations
args(x/y)=args(x)-args(y)
Circle theorem - inclined angle theoremThe Attempt at a Solution
I sketched the circle with major arc.
Radius= using Pythagorus I got the radius as 5 unit^2 .
H=O/sinθ . H=2.5/sin(π/6)
I am stuck on finding the centre point.
[/B]
I don't know how to change it into that form.Ray Vickson said:Your radius is wrong.
Write ##z = x + iy## and express the ratio ##(z-3i)/(z+4)## as ##A(x,y) + i B(x,y)##. How can you get the equation of the curve in terms of the functions ##A(x,y)## and ##B(x,y)##?
The coordinates are (0,3) and (-4,0)mfb said:Can you find one or two points on the circle?
With a complex z and c:$$\frac c z = \frac{cz^*}{zz^*}$$Kajan thana said:I don't know how to change it into that form.
I finally got the answer right and the radius is 5 unit^2. Your way gave me the same answer as well.mfb said:With a complex z and c:$$\frac c z = \frac{cz^*}{zz^*}$$
Here * is the complex conjugation. Now the denominator is real and you can split the fraction into real and imaginary part.