Hyperbola derivation, Pythagorean theorem

[Nazariy]

Hello!

I would like to derive a formula for hyperbola but I struggle to think of a way to explain c in terms of b and a. Here is what I mean, I will use ellipse as an example first:

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If we choose a point that lies exactly on the top minor vertex, then the sum of distances from focii has to be 2a, but the distances from focii to that point are equal, thus each line is a. Height of right angle triangle is b and distance from origin to focii is c. Thus we can express b^2 in terms of a^2 and c^2, which makes it possible to then derive the ellipse equation.

I however have an equation for a hyperbola in terms of a's and c's. I need later on to do a similar manipulation to derive a standard equation for hyperbola. I however cannot seem to be able to define the realtionship, whereas some sources define c^2 = a^2 + b^2 in hyperbola case. Here is the graph (poorly drawn) for hyperbola that I made:

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What we have in particular case is a and height b, we can define that hypotenuse (portion of asymptote) as a^2 + b^2; but I cannot think why that portion length should be c, and that is what people claim it is(or at least what I think they claim it is). If I understand this, I could derive the hyperbola equation.

Thanks

 

[pka]

I however have an equation for a hyperbola in terms of a's and c's. I need later on to do a similar manipulation to derive a standard equation for hyperbola. I however cannot seem to be able to define the realtionship, whereas some sources define c^2 = a^2 + b^2 in hyperbola case. Here is the graph (poorly drawn) for hyperbola that I made:

View attachment 3781​

What we have in particular case is a and height b, we can define that hypotenuse (portion of asymptote) as a^2 + b^2; but I cannot think why that portion length should be c, and that is what people claim it is(or at least what I think they claim it is). If I understand this, I could derive the hyperbola equation.

The hyperbola is the most difficult to derive.
The equation for the above is \(\displaystyle \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) and the slopes of the asymtotes are \(\displaystyle \pm\dfrac{b}{a}\).

The two vertices are \(\displaystyle (a,0)~\&~(-a,0)\) the two focii are on the x-axis but outside \(\displaystyle [-a,a]\).
Most analytic geometery textbooks do not discuss a \(\displaystyle c\) in connection with a hyperbola.

 

[Nazariy]

The hyperbola is the most difficult to derive.
The equation for the above is \(\displaystyle \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\) and the slopes of the asymtotes are \(\displaystyle \pm\dfrac{b}{a}\).

The two vertices are \(\displaystyle (a,0)~\&~(-a,0)\) the two focii are on the x-axis but outside \(\displaystyle [-a,a]\).
Most analytic geometery textbooks do not discuss a \(\displaystyle c\) in connection with a hyperbola.

Yes, I do understand all of that. I use Stanford's single variable calculus concepts and contexts by James Stewart appendix to recap on my coordinate geometry, but they just state that c^2 equals a^2 + b^2 in this case. I cannot figure out why by myself, nor on the internet, been thinking about this for the whole day now.

I can show that for hyperbola:

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And we can obtain the correct hyperbola equation only if I assume that what they say there (about c^2) is correct(which I am sure it is). But as to why that c^2 is the sum of squared a and squared b, I have no idea...

 

[pka]

Yes, I do understand all of that. I use Stanford's single variable calculus concepts and contexts by James Stewart appendix to recap on my coordinate geometry, but they just state that c^2 equals a^2 + b^2 in this case.

I no longer have that book by Stewart. But look at this
They have c so \(\displaystyle c^2=a^2+b^2.\)

 

[Nazariy]

I no longer have that book by Stewart. But look at thisView attachment 3783
They have c so \(\displaystyle c^2=a^2+b^2.\)

Yes, that is what c is supposed to be for it to work out. Let me check. Thank you so much!