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# help hyperbola problem

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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