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A hyperbola passes through the point (sq...

A hyperbola passes through the point `(sqrt2,sqrt3)` and has foci at `(+-2,0)`. Then the tangent to this hyperbola at P also passes through the point:

A

`(3sqrt(2),2sqrt(3))`

B

`(2sqrt(2),3sqrt(3))`

C

`(sqrt(3),sqrt(2))`

D

`(-sqrt(2),sqrt(3))`

Text Solution

Verified by Experts

Let the equation of hyperbola be `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`.
` therefore ae=2 rArr a^(2)e^(2) =4`
`rArr a^(2)+b^(2)=4 rArr b^(2)=4 -a^(2)`
` therefore (x^(2))/(a^(2))-(y^(2))/(4-a^(2))=1`
Since, `(sqrt(2),sqrt(3))` lie on hyperbola.
` :. (2)/(a^(2))-(3)/(4-a^(2))=1`
`rArr 8-2a^(2)-3a^(2)=a^(2)(4-a^(2))`
`rArr 8-5a^(2)=4a^(2)-a^(4)`
`rArr a^(4) - 9a^(2)+8=0`
` rArr (a^(4)-8)(a^(4)-1)=0 rArr a^(4)=8,a^(4)=1`
` therefore a=1`
Now, equation of hyperbola is `(x^(2))/(1)-(y^(2))/(3)=1`.
`therefore` Equation of tangent at `(sqrt(2),sqrt(3))` is given by
`sqrt(2)x-(sqrt(3)y)/(3)=1 rArr sqrt(2)x-(y)/(sqrt(3))=1`
which passes through the point `(2 sqrt(2), 3 sqrt(3))`.
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