Understanding equation, vertices of hyperbola: "Since (sqrt{3}, 4) lies on curve..."

[LaaLa]

Understanding equation, vertices of hyperbola: "Since (sqrt{3}, 4) lies on curve..."

:???:

Usually the vertices have a solid point on the graph which gives you something to work with easily, however, I am slightly puzzled on these equations. (The symbol √ confuses me when put on a graph)
It shows the work to getting the answer, but I don't understand how 'b2' is calculated. Where do all these numbers and factions come from?
Would anyone like to explain to me how this works? It would be greatly appreciated



Here is another example:

 

[tkhunny]

:???:

Usually the vertices have a solid point on the graph which gives you something to work with easily, however, I am slightly puzzled on these equations. (The symbol √ confuses me when put on a graph)
It shows the work to getting the answer, but I don't understand how 'b2' is calculated. Where do all these numbers and factions come from?
Would anyone like to explain to me how this works? It would be greatly appreciated

View attachment 10714

Here is another example:

View attachment 10715

They appear to be the same example.

Small wonder you find this confusing. It appears to me that the author simply pulled \(\displaystyle (\dsqrt{3},4)\) out of the sky. What you need is ANY point on the curve OTHER THAN the two vertices. Are you SURE there is no indication in the text that the point is on the graph? Maybe it was mentioned in a paragraph introducing the problem set?

 

[topsquark]

It looks to me to be that there is a missing diagram. Else how do we know that the x interceps are equal to +/- 1, much less the form of the hyperbola.

-Dan

 

[Steven G]

I think that this is the solution to a problem and not the problem itself. It again amazes me that the poster would not be nice enough to post the problem for us.

3-2 =1, so 16/b^2 =2. Now 16/8 = 2. So b^2 = 8

You can read the definition and derivation of the equation of a hyperbola here:https://courses.lumenlearning.com/i...uation-of-a-hyperbola-centered-at-the-origin/

 

[LaaLa]

Update

The problem you are looking for is on the graph, but I have given you the answer because I do not understand it.
Excuse me if I was inconsiderate or unkind, I never intended to be nasty! I do not know of these things, that's why I am on here asking. Please be patient with me and I will give you your answers you are looking for to help me. Thanks for your support!

I hope this picture will give you a clearer understanding..



Like I said before, I do not understand how this equation is put together. The answer and the working is there but I can't seem to understand how it got there...




I can calculate it any other way but when something like this pops up (...since (2,√3) lies on the curve..), or even this one (...since (√3, 4) lies on the curve..) I can't seem to get my head around working out how to get the answer for b^2. It doesn't say anywhere how these equations are understood. I searched on google and even youtube...

 

[tkhunny]

You are shown in the diagramme that \(\displaystyle (2,\sqrt{3})\) is on the graph. Substitute \(\displaystyle x=2\;and\;y=\sqrt{3}\) into your equation.

 

[Steven G]

View attachment 10720

I answered your questions in my earlier post. What did you not understand?

 

[sinx]

The problem you are looking for is on the graph, but I have given you the answer because I do not understand it.
Excuse me if I was inconsiderate or unkind, I never intended to be nasty! I do not know of these things, that's why I am on here asking. Please be patient with me and I will give you your answers you are looking for to help me. Thanks for your support!

I hope this picture will give you a clearer understanding..
View attachment 10718


Like I said before, I do not understand how this equation is put together. The answer and the working is there but I can't seem to understand how it got there...
View attachment 10716

View attachment 10720

I can calculate it any other way but when something like this pops up (...since (2,√3) lies on the curve..), or even this one (...since (√3, 4) lies on the curve..) I can't seem to get my head around working out how to get the answer for b^2. It doesn't say anywhere how these equations are understood. I searched on google and even youtube...

general eqn for hyperbola with origin (0,0) is; x²/a²-y²/b²=1
where a=x intercept
and b=co-vertices
so the specific eqn for this hyperbola is; x²-y²/8=1
because it intersects x at -1,1, and has covertices=+-sqrt8

 

[JeffM]

The problem you are looking for is on the graph, but I have given you the answer because I do not understand it.
Excuse me if I was inconsiderate or unkind, I never intended to be nasty! I do not know of these things, that's why I am on here asking. Please be patient with me and I will give you your answers you are looking for to help me. Thanks for your support!

I hope this picture will give you a clearer understanding..
View attachment 10718


Like I said before, I do not understand how this equation is put together. The answer and the working is there but I can't seem to understand how it got there...
View attachment 10716

View attachment 10720

I can calculate it any other way but when something like this pops up (...since (2,√3) lies on the curve..), or even this one (...since (√3, 4) lies on the curve..) I can't seem to get my head around working out how to get the answer for b^2. It doesn't say anywhere how these equations are understood. I searched on google and even youtube...

To get quicker and less annoyed help in the future, please read

https://www.freemathhelp.com/forum/threads/112086-Guidelines-Summary?p=433156&viewfull=1#post433156

The major problem with your first post was that you did not give a complete and exact statement of the problem that you were supposed to answer.

Jomo's first post was a brief explanation of the answer to a problem that he had to guess.

So your second post is where you should have started. However, it is still not clear to me at least what you are confused about.

The given explanation expects that you will recognize that the graph is for a hyperbola centered on (0, 0). Is that what confuses you?

The entire family of such curves can be described by equations of the form

\(\displaystyle \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1.\)

The given explanation expects that you will know that formula. Is that what confuses you?

With such a hyperbola, you can determine what \(\displaystyle a^2\) is from either of the x-intercepts. At an intercept of the x-axis, y = 0 so \(\displaystyle b^2\) effectively plays no role. The x-intercepts are (-1, 0) and (1, 0). Whichever we choose x^2 = 1 and y^2 = 0. Put those values into the general equation.

\(\displaystyle \dfrac{1}{a^2} - \dfrac{0}{b^2} = 1 \implies \dfrac{1}{a^2} = 1 \implies a^2 = 1.\)

Any questions about that? So with respect to this specific hyperbola we can say

\(\displaystyle \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \text { and } a^2 = 1 \implies \dfrac{x^2}{1} - \dfrac{y^2}{b^2} = 1 \implies x^2 - \dfrac{y^2}{b^2} = 1.\)

Now if we knew the coordinates of just one other point on this hyperbola, we could calculate the \(\displaystyle b^2\) for this specific hyperbola.

But the problem gives us such a point, namely by telling us that the point \(\displaystyle \sqrt{3},\ 4)\) lies on this specific hyperbola. In other words, those specific values of x and y satisfy the equation.So

\(\displaystyle x^2 - \dfrac{y^2}{b^2} = 1, \ x = \sqrt{3} \text { and } y = 4 \implies\)

\(\displaystyle (\sqrt{3})^2 - \dfrac{4^2}{b^2} = 1 \implies 3 - \dfrac{16}{b^2} = 1 \implies \dfrac{16}{b^2} = 3 - 1 \implies \)

\(\displaystyle \dfrac{16}{b^2} = 2 \implies 2b^2 = 16 \implies b^2 = 8.\)

Therefore, this hyperbola has the equation \(\displaystyle x^2 - \dfrac{y^2}{8} = 1\).

If you look carefully, you will see that several posts have given a condensed version of this answer.

Are you still confused?

 

[LaaLa]

Understand equation, vertices of hyperbola: "Since (√3, 4) lies on the curve..."

To get quicker and less annoyed help in the future, please read

https://www.freemathhelp.com/forum/threads/112086-Guidelines-Summary?p=433156&viewfull=1#post433156

The major problem with your first post was that you did not give a complete and exact statement of the problem that you were supposed to answer.

Jomo's first post was a brief explanation of the answer to a problem that he had to guess.

So your second post is where you should have started. However, it is still not clear to me at least what you are confused about.

The given explanation expects that you will recognize that the graph is for a hyperbola centered on (0, 0). Is that what confuses you?

The entire family of such curves can be described by equations of the form

\(\displaystyle \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1.\)

The given explanation expects that you will know that formula. Is that what confuses you?

With such a hyperbola, you can determine what \(\displaystyle a^2\) is from either of the x-intercepts. At an intercept of the x-axis, y = 0 so \(\displaystyle b^2\) effectively plays no role. The x-intercepts are (-1, 0) and (1, 0). Whichever we choose x^2 = 1 and y^2 = 0. Put those values into the general equation.

\(\displaystyle \dfrac{1}{a^2} - \dfrac{0}{b^2} = 1 \implies \dfrac{1}{a^2} = 1 \implies a^2 = 1.\)

Any questions about that? So with respect to this specific hyperbola we can say

\(\displaystyle \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 \text { and } a^2 = 1 \implies \dfrac{x^2}{1} - \dfrac{y^2}{b^2} = 1 \implies x^2 - \dfrac{y^2}{b^2} = 1.\)

Now if we knew the coordinates of just one other point on this hyperbola, we could calculate the \(\displaystyle b^2\) for this specific hyperbola.

But the problem gives us such a point, namely by telling us that the point \(\displaystyle \sqrt{3},\ 4)\) lies on this specific hyperbola. In other words, those specific values of x and y satisfy the equation.So

\(\displaystyle x^2 - \dfrac{y^2}{b^2} = 1, \ x = \sqrt{3} \text { and } y = 4 \implies\)

\(\displaystyle (\sqrt{3})^2 - \dfrac{4^2}{b^2} = 1 \implies 3 - \dfrac{16}{b^2} = 1 \implies \dfrac{16}{b^2} = 3 - 1 \implies \)

\(\displaystyle \dfrac{16}{b^2} = 2 \implies 2b^2 = 16 \implies b^2 = 8.\)

Therefore, this hyperbola has the equation \(\displaystyle x^2 - \dfrac{y^2}{8} = 1\).

If you look carefully, you will see that several posts have given a condensed version of this answer.

Are you still confused?

I assumed everyone knew what (√3,4) was. To me, it could be ANY point or number given to solve the equation. I never really cared where the number (√3,4) had come from and solving the equation was more important to me. I also assumed that everyone else never cared about those details either, but I guess you can never be sure about things until you see it with your own eyes right? My mistake... I still believe that where the number comes from doesn't matter... Each to their own! It would be easier for me to understand and solve the equation if the point given was (3,4) instead of that strange symbol in the way (√). I had never dealt with those symbols before and the study never explained how to use them. I’ve come across a few of them so far and it wasn’t till later in the study that it said I needed a calculator to get the numbers. Then I had to youtube to find out how to work the calculator to get my answers. It had taken me a whole week to figure this all out. This study is all back to front!

I also had to grab a sheet and write a summary of all the fractions and decimals down on a piece of paper for those harder equations that have the √ symbol. It saves me wondering where all these fractions come from. It’s because the calculator only gives a decimal figure, but it can’t translate to a fraction. I was so confused on all of this before but now it makes sense!

JeffM thank you for the excellent post reply.
It was very thorough and I have found your explanation to be very useful! I will save this for future reference Your a life saver!