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# Hyperbola and Foci? :(

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The graph of $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

has its foci at $(0,\pm 4),$

while the graph of $\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$

has its foci at  $(\pm 6,0).$

Find a and b

Jan 13, 2021

#1
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All the info you need is contained in these 2 diagrams.

Incorrect graph has been deleted :  sorry about that.

Jan 13, 2021
edited by Melody  Jan 13, 2021
edited by Melody  Jan 16, 2021
#2
+1

deleted

Guest Jan 14, 2021
edited by Melody  Jan 16, 2021
#3
+112816
+1

deleted

Melody  Jan 14, 2021
edited by Melody  Jan 16, 2021
#4
+1

From the circle I am getting

16 = a2 - b2

and from the hyperbola I am getting

36 = a2 + b2

solving these 2 equations gets

20 = 2b2

b = $$\sqrt{10}$$

a = $$\sqrt{26}$$

however when I try to graph it, it looks nothing like your graph

https://www.desmos.com/calculator/qeszzqvt5h

Guest Jan 15, 2021
#5
+112816
+1

The foci of an ellipse have to be inside the ellipse.  (0, 4) and (0, -4) are not in your ellipse.

Melody  Jan 15, 2021
#6
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deleted  - it was wrong

Jan 15, 2021
edited by Melody  Jan 16, 2021
#7
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Can you show me how you got those numbers?

I am still confused but I can understand more than before.

Guest Jan 15, 2021
#8
+112816
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I am glad you have been persistent.

Keep it up.

I don't have my earlier working any more.  It beats me where I got those numbers from.  Sorry.

Let me start over.

The equation of an ellipse is    $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

focal length c = 4

If the major axis is horizonal then a > b

If the major axis is vertical then a < b

Since this one has a major axis of  y=0 it is vertical so  a is smaller than b

The focal length is c where

$$c^2=|a^2-b^2|$$

since a < b,

$$c^2=b^2-a^2\\ 16=b^2-a^2\\$$

For the hyperbola

$$c^2=a^2+b^2\\ 36=a^2+b^2\\$$

solve them simultaneously and you get   $$b=\sqrt{26}\qquad a=\sqrt{10}$$

Here is the graph

https://www.desmos.com/calculator/4wam1hbltm

Jan 16, 2021
edited by Melody  Jan 16, 2021
#9
+1

I finally understand now!

Sorry for being stubborn and not understanding quicker

I appreciate the time you took to help me

Keep it up Melody

Guest Jan 16, 2021
#10
+112816
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It was a bit hard for you to understand when nothing I had written made any sense :)

Melody  Jan 16, 2021