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If it is possible to draw the tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` having slope 2, then find its range of eccentricity.

Text Solution

Verified by Experts

For the hyperbola
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
the tangent having slope m is `y=mx pm sqrt(a^(2)m^(2)-b^(2))`.
The tangent having slope 2 is `y=2x pm sqrt(4a^(2)-b^(2))`, which is real
`4a^(2)-b^(2)ge0`
`"or "(b^(2))/(a^(2))le4`
`"or "e^(2)-1le4`
`"or "e^(2)le5`
`"or "1lteltsqrt5`
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