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If the hyperbola (x^(2))/(a^(2))-(y^(2))...

If the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))` passes through the points `(3sqrt2, 2)` and `(6, 2sqrt(3))`, then find the value of e i.e., eccentricity of the given hyperbola.

Text Solution

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It is given that the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))` is passing through the points `(3sqrt(2), 2) and (6, 2sqrt(3))`.
`therefore` These points will satisfy the given equation i.e., `((3sqrt(2))^(2))/(a^(2))-((2)^(2))/(b^(2))=1 and ((6)^(2))/(a^(2))-((2sqrt(3))^(2))/(b^(2))=1`
`rArr (18)/(a^(2)) - (4)/(b^(2)) -1 and (36)/(a^(2)) -(12)/(b^(2)) =1`
`rArr 18b^(2) - 4a^(2) = a^(2)b^(2) and 36b^(2) - 12a^(2)b^(2)" "...(1)`
Solving the two equations in (1), we get :
`a^(2) = 9 and b^(2) = 4`
Now, we have, `b^(2) = a^(2)(e^(2) -1)`
i.e., `4 = 9(e^(2) -1)`
i.e, `e^(2) = 1+ (4)/(9)`
` = (13)/(9)`
`therefore e = (sqrt(13))/(3)`
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