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Consider a hyperbola: ((x-7)^(2))/(a^2) ...

Consider a hyperbola: `((x-7)^(2))/(a^2) -((y+3)^(2))/(b^(2)) =1`. The line `3x - 2y - 25 =0`, which is not a tangent, intersect the hyperbola at `H ((11)/(3),-7)` only. A variable point `P(alpha +7, alpha^(2)-4) AA alpha in R` exists in the plane of the given hyperbola.
Which of the following are not the values of `alpha` for which two tangents can be drawn one to each branch of the given hyperbola is

A

`(2,oo)`

B

`(-oo,-2)`

C

`(-(1)/(2),(1)/(2))`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
Equation of asymptotes to hyperbola is `((x-7)^(2))/(a^(2)) - ((y+3)^(2))/(b^(2)) =0` or `(b^(2)(x-7)^(2))/(a^(2)) -(y+3)^(2) =0`
or `(9(x-7)^(2))/(4) -(y+3)^(2) =0`
Now point `P(alpha + 7, alpha^(2)-4)` is such that two tangents can be drawn one to each branch of the given hyperbola, then `(9(alpha+7-7)^(2))/(4) - (alpha^(2)-1)^(2) lt 0`
`rArr (alpha^(2)-1)^(2) gt (9alpha^(2))/(4)`
`rArr (2alpha^(2)-3alpha -2) (2alpha^(2)+3alpha-2) gt 0`
`rArr (2alpha +1) (alpha-2) (2alpha-1) (alpha+2) gt 0`
`rArr alpha in (-oo,-2) uu (-1//2,1//2) uu(2,oo)`
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