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If tangent of y^(2)=x at (alpha,beta), w...

If tangent of `y^(2)=x` at `(alpha,beta)`, where `betagt0` is also a tangent of ellipse `x^(1)2y^(2)=1` then value of `alpha` is

A

`2 sqrt(2) + 1`

B

`sqrt(2) + 1`

C

`2 sqrt(2) - 1`

D

`sqrt(2) - 1`

Text Solution

Verified by Experts

The correct Answer is:
B
Let a point lying on the parabola `y^(2) = x` is `(beta^(2), beta)`
Equation of tangent at `(beta^(2), beta)` to the parabola `y^(2) = x` is `y beta = (X + beta^(2))/(2)`
`implies y = (x)/(2 beta) + (beta)/(2)`
If this line also touches the ellipse `(x^(2))/(1) + (y^(2))/(1//2) =` then `C^(2) = a^(2) m^(2) + b^(2)`
`(beta^(2))/(4) = 1 . (1)/(4 beta^(2)) + (1)/(2)`
`implies beta^(4) = 1 + 2 beta^(2) implies beta^(4) - 2 beta^(2) - 1 = 0`
`(beta^(2) - 1)^(2) = 2 implies beta^(2) - 1 = +- sqrt(2)`
`beta^(2) = 1 +- sqrt(2)` (ive sign rejected)
`implies beta^(2) = 1 + sqrt(2) implies alpha = 1 + sqrt(2)`
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