1.

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

Answer»

`(2,oo)`
`(-oo,-2)`
`(-(1)/(2),(1)/(2))`
None of these

Solution :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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