Relation between a circle and line in complex plane

[Darshit Sharma]

TL;DR

Distance of line from center

I was reading a book on complex numbers where I stumbled upon this article.

I was following it when I found some error.

The distance between the center of circle and line should be mod(z star - z1) right?
Z0 is just any random pt on line.

They have continued this fallacy in example 3.10 also.


Am I correct?


I have added zoomed in images too

 

[PeroK]

The attachments are too small to read.

 

[Darshit Sharma]

The attachments are too small to read.

I've added more zoomed in versions sir

 

[PeroK]

I've added more zoomed in versions sir

How does the author define $\bar z$?

 

[PeroK]

PS if $\bar z$ is the complex conjugate, then the whole thing looks wrong.

 

[Darshit Sharma]

PS if $\bar z$ is the complex conjugate, then the whole thing looks wrong.

Yes and eta is the complex slope

 

[Darshit Sharma]

PS if $\bar z$ is the complex conjugate, then the whole thing looks wrong.

Here's link to books pdf sir https://matematickitalent.mk/uploads/books/inpRRqguAkijOWpiXCW4aA.pdf

 

[PeroK]

Yes and eta is the complex slope

The equation $z = \bar z + 3i$ reduces to $y = \frac 3 2$, where $z = x + yi$.

And, the equation $|z + 4 - 2i| = 3$ is the circle radius $3$, centred on $-4 + 2i$.

Clearly, that line intersects the circle at two points, as it passes the centre at a distance of $\frac 1 2$.

PS using this geometric approach, you can see that the point on the line closest to the centre of the circle is $-4 + \frac 3 2 i$.

The book's conclusion is truly bizarre!

 

[Darshit Sharma]

Yes sir....they have used their previously derived eqns using z0 as the center of the circle lol.......Thanks sir I just wanted to verify. And sir can you suggest some books on complex numbers please

 

[PeroK]

Yes sir....they have used their previously derived eqns using z0 as the center of the circle lol.......Thanks sir I just wanted to verify. And sir can you suggest some books on complex numbers please

I don't have a book on complex numbers.

 

[FactChecker]

It's probably just a typo. For instance, its conclusion would have worked for the line $z=\bar{z} -3i$. Or the center of the circle could be shifted.
Proof reading a book is an endless, thankless job. You might inform the author of the mistake.

 

[FactChecker]

can you suggest some books on complex numbers please

I have never seen a book like that on complex numbers. There are a multitude of books on complex analysis but without nearly that much detail on the geometry of complex numbers.

 

[PeroK]

It's probably just a typo. For instance, its conclusion would have worked for the line $z=\bar{z} -3i$. Or the center of the circle could be shifted.
Proof reading a book is an endless, thankless job. You might inform the author of the mistake.

The authors used $z_1$ for the centre of the circle, but when it came to calculate the distance to the centre of the circle, they used $z_0$.

That's an easy mistake to make. But, if they had checked the answer to example 3.10, they would have spotted the error in that and the formulas in the next section.

 

[WWGD]

TL;DR Summary: Distance of line from center

I was reading a book on complex numbers where I stumbled upon this article.

I was following it when I found some error.

View attachment 355154

The distance between the center of circle and line should be mod(z star - z1) right?
Z0 is just any random pt on line.

They have continued this fallacy in example 3.10 also.


Am I correct?


I have added zoomed in images tooView attachment 355155View attachment 355157View attachment 355158

I assume this refers to the _shortest_ distance between the two. So you consider a line from the center of the circle that is perpendicular to the line, then find the distance between the center of the circle and point of intersection with the line given. Have you tried this?

 

[Darshit Sharma]

I assume this refers to the _shortest_ distance between the two. So you consider a line from the center of the circle that is perpendicular to the line, then find the distance between the center of the circle and point of intersection with the line given. Have you tried this?

Yup the perpendicular dist of line from center.....which they have done wrong in the book.

 

[Darshit Sharma]

It's probably just a typo. For instance, its conclusion would have worked for the line $z=\bar{z} -3i$. Or the center of the circle could be shifted.
Proof reading a book is an endless, thankless job. You might inform the author of the mistake.

Sure sir

 

[MidgetDwarf]

Yes sir....they have used their previously derived eqns using z0 as the center of the circle lol.......Thanks sir I just wanted to verify. And sir can you suggest some books on complex numbers please

One of the easiest books to learn from, is Churchill and Brown.

A similar book is the one by Zill. I own a copy of Zill, never read it, but someone I mentored was taking Complex Analysis [undergrad], and I let them borrow 5 books.

They chose Zill.

For a more advance treatment. There is the book by Lang.

Are you doing pure math or more for physics/engineering. What level of mathematics are you at.