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A hyperbola passes through the point P(s...

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

A

(a) `(-sqrt2,-sqrt3)`

B

(b) `(3sqrt2,2sqrt3)`

C

(c) `(2sqrt2,3sqrt3)`

D

(d) `(sqrt3,sqrt2)`

Text Solution

Verified by Experts

The correct Answer is:
C
Equation of hyperbola is `(X^(2))/(a^(2))-(y^(2))/(b^(2))=1`.
Foci are `(pm2,0)`. So , ae = 2.
Now, `b^(2)=a^(2)(e^(2)-1)`
`therefore" "a^(2)+b^(2)=4" (1)"`
Given hyperbola passes through `(sqrt2,sqrt3).`
`therefore" "(2)/(a^(2))-(3)/(b^(2))=1" (2)"`
One solving (1) and (2), we get
`a^(2)=1 and b^(2)=3`
So, equation of the hyperbola is `(x^(2))/(1)-(y^(2))/(3)=1`.
Hence, equation of tangent at `P(sqrt2,sqrt3)" is "(sqrt2x)/(1)-(sqrt3y)/(3)=1.`
which passes through the point `(2sqrt2,3sqrt3)`.
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