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 $h + k + a + b.$

Guest Oct 2, 2020

#3**+1 **

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.

Melody Oct 6, 2020