Given that hyperbola `xy-2y + 3x =k` is tangent to the ellipse `3x^(2)-12x+2y^(2)+12y+6=0` at two points A and B On the bases of above answer the following
Length of chord AB of ellipse is

Options

A. `sqrt(22)`
B. `sqrt(30)`
C. `sqrt(40)`
D. `5`

Answer

C

Explanation

`(x-2)(y+3)=k-6`
`(x-2)^(2)/(8)+(y+3)^(2)/(12)=1`
`x-2=X,y+3-Y`
`Xy=k-6" " …..(1)`

`X^(2)/(8)+Y^(2)/(12)=1" " ….(2)`
At the point A and B, slopes of tangent to both curves must be equal
`impliesy^(')=(beta)/(alpha)=(-3alpha)/(2beta)`
`2beta^(2)=3alpha^(2)`
since `(alpha,beta)` satisfies equation (ii)
`(alpha^(2))/(8)+beta^(2)/(12)=1`
solving `alpha =pm2,beta= pm sqrt(6)`
`alpha,beta in (2,sqrt(6))(-2,-sqrt(6))`
`alpha, beta in (2,-sqrt(6))(-2,sqrt(6))`
`k=6+2sqrt(6)`
`k=6-2sqrt(6)`
coordinates of tangency
coordinates of tangency
`(4,sqrt(6)-3)(0,-sqrt(6)-3)`
`(4,-sqrt(6)-3)(0,sqrt(6)-3)`