A hyperbola passes through the point `P(sqrt2,sqrt3)` and has foci at `(pm2,0)`. Then the tangent to this hyperbola at P also passes through the point

Options

A. `(sqrt(3),sqrt(2))`
B. `(-sqrt(2),-sqrt(3))`
C. `(3sqrt(2),2sqrt(3))`
D. `(2sqrt(2),3sqrt(3))`

Explanation

Let the equation of the hyperbola be
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`……`(i)`
It pasess through `P(sqrt(2),sqrt(3))` and has foci at `S(2,0)` and `S'(-2,0)`
`:.ae=2` and `SP+S'P=2a`
`impliesae=2 & |sqrt((2-sqrt(2))^(2)+(0-sqrt3)^(2))-sqrt((-2-sqrt(2))^(2)+(0-sqrt(3))^(2))|=2a`
`impliesae=2` and `|sqrt(9-4sqrt(2))-sqrt(9+4sqrt(2))|=2a`
`impliesae=2` and `|(2sqrt(2)-1)-(2sqrt(2)+1)|=2a`
`impliesae=2` and `a=1impliesa=1` and `e=2`
`:.b^(2)=a^(2)(e^(2)-1)impliesb^(2)=3`.
So the equation of the hyperbola is `x^(2)-(y^(2))/(3)=1` or `3x^(2)-y^(2)=3`.
The equation of tangent `P(sqrt(2),sqrt(3))` is `3sqrt(2)x-sqrt(3)y=3`.
Clearly it passes through `(2sqrt(2),3sqrt(3))`.