The equation of a hyperbola is x^2/24^2 - y^2/ (10)^2= 1.
We have given that, The equation x^2/24^2 - y^2/ (b)^2= 1 represents a hyperbola centered at the origin with a directrix of x = 576/26.
We have to find the value of b.
[tex]\frac{(x-h)^2}{a^2} -\frac{(y-k)^2}{b^2} =1[/tex]
We have given that,
[tex]x^2/24^2 - y^2/ (b)^2= 1 \implies \frac{(x-0)^2}{24} -\frac{(y-0)^2}{b^2} =1[/tex]
a=24,h=0 and k=0
Now equation of the directrix
x=a^2/c...(1)
and we know x=576/26...(2)
Therefore from 1 and 2 we get
24^2/c=576/26.
isolate the c so we get,
C=26
C= center of focii
[tex]c=\sqrt{a^2+b^2} \\c^2=a^2+b^2\\b^2=c^2-a^2\\b=10[/tex]
So we get the value of b is 10.
Therefore the equation of a hyperbola is x^2/24^2 - y^2/ (10)^2= 1.
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