Question

Find the eccentricity of the hyperbola given by equations `x=(e^t+e^(-1))/2a n dy=(e^t-e^(-1))/3,t in Rdot`

Answer 1

Given equations
`x=(e^(t)-e^(-t))/(2)and y=(e^(t)-e^(-t))/(3),`
`"or "2x=e^(t)+e^(-t) and 3y=e^(t)-e^(-t)`
Squaring and subtracting, we get
`4x^(2)-9y^(2)=4`
`"or "(x^(2))/(1)-(y^(2))/(4//9)=1`
Now, `b^(2)=a^(2)(e^(2)-1)`
`therefore" "e^(2)=(4)/(9)+1=(13)/(9)`
`"or "e=(sqrt(13))/(3)`