## conic sections – How can one verify the equivalence between the geometric and algebraic definitions of an ellipse/htperbola?

I was recently introduced to the world of conic sections, and I came across two different “definitions” for an ellipse and a hyperbola.

I was wondering if there was an easy way to prove that they happen to be “equivalent”.

First, it was established that:

Given a line $$r$$, a point $$F$$ not in $$r$$, and $$e in mathbb R^+_*$$, an ellipse/hyperbola is the locus of the points $$P in mathbb R^2$$ such that $$overline{PF} = eoverline{Pr}$$ (where $$e<1$$ for the ellipse and $$e>1$$ for the hyperbola)

What I would like to verify is that the set of points $$bigg{(x,y) in mathbb R^2: dfrac{x^2}{a^2} pm dfrac{y^2}{b^2} =1bigg}$$ satisfy the definition given.

Any thoughts?

## Ella – Multipurpose Shopify Sections Theme

Ella – Multipurpose Shopify Sections Theme – Ella – Multipurpose Shopify Sections Theme

View attachment 32586

[IMG alt="Full Theme…

.

## javascript – Show contents in a book chapter and sections in a highly normalized repository

I want to present books that has chapters and sections, and contents in each book, chapter and section.

This is my schema:

``````type BookId = string
type ChapterId = string
type SectionId = string
type ContentId = string

export type SourceId =
| ChapterId
| SectionId
| BookId

type Book = {
id: BookId,
name: string,
}

type Chapter = {
id: ChapterId,
bookId: BookId,
name: string
}
type Section = {
id: SectionId,
chapterId: ChapterId,
name: string
}

type Content = {
id: ContentId,
sourceId: SourceId,
name: string,
content: string
}
``````

This is how I query all the books, chapters and sections:

`````` list() {
return this.bookRepo.all().then(books =>
Promise.all(books.map(book =>
Promise.all((
this.bookRepo.contents(book.id),
this.bookRepo.chapters(book.id)
)).then(((contents, chapters)) =>
Promise.all(chapters.map(chapter =>
Promise.all((
this.bookRepo.contents(chapter.id),
this.bookRepo.sections(chapter.id)
)).then(((contents, sections)) =>
Promise.all(sections.map(section =>
this.bookRepo.contents(section.id).then(contents => (
section, contents
))))
.then(sections => (chapter, contents, sections)))))
.then(chapters => (book, contents, chapters))))));
}
``````

I just render as json at the moment, but the query code seems hard to maintain.

## Coordinate geometry conic sections

A square ABCD of unit side is drawn along with a circle C1 of unit radius centered at A. Another circle C2 is drawn such that it touches C1 and AB and AD being tangents to it. If a tangent at C2 from point C touches the side AB at E then the length of EB is x+y((3)^(1/2)) then x+y=?​

## at.algebraic topology – Does homeomorphism between cones imply homeomorphism between sections

The answer is no. Let $$X$$, $$Y$$ be any two smoothly h-cobordant closed manifolds of dimension $$ge 4$$ that are non-homeomorphic. For example, we can take $$X=S^2times L(7,1)$$ and $$Y=S^2times L(7,2)$$ where as usual $$L(p,q)$$ is a 3-dimensional lens space. By the weak h-cobordism theorem the interior of the h-cobordism is diffeomorphic to $$Xtimesmathbb R$$ and also to $$Ytimes mathbb R$$. The interior is two-ended, and we can compactify it by adding two points (= the ends). The “identity” homeomorphism $$Xtimesmathbb Rto Ytimes mathbb R$$ extends to
a homeomorphism of the compactifications that takes ends to ends. Removing the points corresponding to one end gives a homeomorphism from the cone over $$X$$ to the cone over $$Y$$ that preserves the cone point.

Conversely, if $$X$$, $$Y$$ are closed manifolds and $$Xtimesmathbb R$$ is diffeomorphic to $$Ytimes mathbb R$$, then $$X$$, $$Y$$ are h-cobordant.

The same trick of course works when $$X$$, $$Y$$ are compact non-homeomorphic spaces and $$Xtimesmathbb R$$ is homeomorphic to $$Ytimes mathbb R$$.

## Cognito Forms: Query across repeating sections by ItemNumber

Is there a way to lookup by ItemNumber across repeating section?

This can really help the design when you have large sections that are bound by the number of items but should be logically separate in the design and work-flow. For example, sections get filled out at different times in the work-flow. Think of it as a one-to-one join.

Design:

Enter two items into the form:

The next section should show the items separate, however, it lists both items in both fields:

See: “Referencing repeating data” here:
https://www.cognitoforms.com/support/51/calculations/repeating-sectionstables

`=Form.RepeatingSection1.Where(ItemNumber = ItemNumber).Select(UserInput)`

This could be a cognito limitation / bug. I can tell that the first ItemNumber in the Where function definitely references any field in RepeatingSection1 because of the drop-down autocomplete list so we would expect that the the second ItemNumber would properly reference the current section.

## Can one edit the sections in a Google form after sending it?

Yes you can.

First of all make a copy of your form(a test form), link it to a new answer sheet and submit couple of test replies.
Following that, make the changes you want and submit again some replies.

If you notice no changes to your answer sheet, continue by making the same changes to the original form. Problem solved 🙂

Important note
If I were you, I would first make a note -by checking the already existing timestamps- of when the least activity is registered and take the following steps during a passive period.

1. Change the logic of your sections

Following the above steps a new tab will be created in your answer sheet where new submissions will be recorded.
To have results from both tabs together, create a third tab where by using formulas you can import date both previous tabs.

## How to retrieve sections of a form I accidently deleted (including the surveyors responses)?

First time using Google Forms and I was so impressed! It was doing everything I needed for research on a book I am writing. However, Thursday night… on my phone looking at some responses I accidentally deleted the questions (and the responses). I could not find them anywhere and re-created the document but I am now missing my first 40 responses. The updated survey still includes the #40 replies at the top but the answers are not there. Can it be recovered?

## google drive – How to retrieve sections of a survey I accidently deleted (including the surveyors responses)?

First time using Google Surveys and I was so impressed! It was doing everything I needed for research on a book I am writing. However, Thursday night… on my phone looking at some responses I accidentally deleted the questions (and the responses). I could not find them anywhere and re-created the document but I am now missing my first 40 responses. The updated survey still includes the #40 replies at the top but the answers are not there. Can it be recovered? Please help!

## Microsoft Onenote collapse all sections

When I open a Onenote App on windows 10, 13 sections are opened and remaining sessions are collapsed. I would like all of them to be closed /collapsed except the one I am working on. So only one open and rest closed. Is it possible?