Question 723547
Find the equation of a hyperbola with center at (0,0) and passing through points (-1,-2) and (3,4).
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Standard form of hyperbola with vertical transverse axis with center at origin:
{{{y^2/a^2-x^2/b^2=1}}}
use two given points to set up two equations to solve a and b.
..
4/a^2-1/b^2=1
16/a^2-9/b^2=1
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16/a^2-4/b^2=4
16/a^2-9/b^2=1
subtract
5/b^2=3
b^2=5/3
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4/a^2-3/5=1
4/a^2=8/5
a^2=20/8=5/2
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Equation:{{{y^2/(5/2)-x^2/(5/3)=1}}}
This is an equation of a hyperbola with vertical transverse axis.
note: I tried with a horizontal transverse axis at first but that did not come out right.