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Let $F_1 = \left( -3, 1 - \frac{\sqrt{5}}{4} \right)$ and $F_ 2= \left( -3, 1 + \frac{\sqrt{5}}{4} \right).$ Then the set of points $P$ such that \[|PF_1 - PF_2| = 1\]form a hyperbola. The equation of this hyperbola can be written as \[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1,\]where $a, b > 0.$ Find $h + k + a + b.$

 Oct 2, 2020
 #1
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h + k + a + b = -8/3.

 Oct 3, 2020
 #2
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that is incorrect please more help!

Guest Oct 4, 2020
 #3
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Draw it!

 

The centre is obviously (-3,1)   so this gives h and k 

and it is going to have a vertical axis.

 

\(\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\)

 

 

c is the distance from the  centre to a focus so          \(c=\frac{\sqrt5}{4}\)

 

\(|PF_1-PF_2|=1=2a\\ so\;\;\;a=0.5\)

 

Then you have     \(c^2=a^2+b^2\)

 

so you can find b

 

Not so difficult after all.

 Oct 6, 2020
 #4
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Oh!

Thank you, I figured out the answer 😁

Guest Oct 6, 2020

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