1.

If tangent is drawn at (x_(1), y_(1)) on the hyperbola (x^(2))/4-(y^(2))/3=1 intersect the x axis at C(alpha, 0) and y axis at D(0,beta) then int_(2)^(4sqrt(2)) (alpha)/(beta) dx_(1) is

Answer»

`(3pi)/8`
`(-pi)/4`
`(-pi)/8`
`(pi)/8`

Solution :`:' alpha=4/(x_(1)` and `beta=(-3)/(y_(1))=(-4)/(sqrt(x_(1)^(2)-4))`
`:. int_(2)^(4sqrt(2))(4//x_(1))/(-4//sqrt(x_(1)^(2)-4)) dx_(1)=- int_(2)^(4sqrt(2)) (dx_(1))/(x_(1)sqrt(x_(1)^(2)-4))` (Put `x_(1)=2sec theta, dx_(1)=2sec theta tan theta d theta`)
`=-int_(0)^(2)((pi)/4 2sec theta tan theta d theta)/(2 sec theta.2tan theta)=-(pi)/8`


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