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A tangent drawn to hyperbola (x^(2))/(a^...

A tangent drawn to hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =1`at `P((pi)/(6))` forms a triangle of area `3a^(2)` square units, with coordinate axes, then the squae of its eccentricity is equal to

A

15

B

16

C

17

D

18

Text Solution

Verified by Experts

The correct Answer is:
C
The point `P ((pi)/(6))` is `(asec.(pi)/(6),b tan.(pi)/(6))`, i.e., `P((2a)/(sqrt(3)),(b)/(sqrt(3)))`
`:.` Equation of tangent at P is `(x)/((sqrt(3)a)/(2))-(y)/(sqrt(3)b) =1`
`:.` Area of the triangle `=(1)/(2) xx (sqrt(3)a)/(2) xx sqrt(3) b = 3a^(2)`
`:. (b)/(a) = 4`
`:. e^(2) = 1 + (b^(2))/(a^(2)) = 17`
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