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If two distinct tangents can be drawn from the point `(alpha, alpha+1)` on different branches of the hyperbola `(x^(2))/(9)-(y^(2))/(16)=1`, then find the values of `alpha`.

Text Solution

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Point `P(alpha, alpha+1)` lies on the line.
`y=x+1`
It if given that form point P two tangents can be drawn to different branches of hyperbola `(x^(2))/(9)-(y^(2))/(16)=1`.
So, point P lies in one of the four regions formed by asymptotes where branch of hyperbola does not lie.

Equation of asymptotes are
`4x-2y=0`
`"and "4x+3y=0`
If point P lies on the line (2), then `4alpha-3(alpha+1)or alpha=3`.
If point P lies on the line (3), then `4alpha+3(alpha+1)=0or alpha=-3//7.`
Thus, `alphain(-3//7,3)`
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