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

Options

A. `sqrt(2)+1`
B. `sqrt(2)-1`
C. `2 sqrt(2)+1`
D. `2sqrt(2)-1`

Answer

A

Explanation

Since the point `( alpha , beta ) `is on the parabola `y^(2)= x, ` so
Now , equation of tangent at point `(alpha , beta)` to the parabola `y^(2)= x, ` is ` T=o `
`implies y beta =(1) /(2) (x+alpha)`
`[:' ` equation of the tangent to the parabola `y^(2)=4ax` at a point `(x_(1) ,y_(1))` is given by `YY_(1)= 2a (x+x_(1))]`
`implies 2 yBeta = (1) /(2) ( x+a)`
`[:' ` equation of the tangent to the parabola `y^(2)= 4ax` at a point `(x_(1) ,y_(1) ) ` is given by `Y Y_(1) = 2a ( x+x_(1)]`
` 2 y beta = x+ beta ^(2) `
`implies y= (x)/(2 beta)+(beta)/(2) `
since . line (ii) is also a tangent of the ellipse
`x^(2)+2y^(2)=1`
`therefore ((beta)/(2))^(2)=(1)^(2)((1)/(2 beta))^(2)+(1)/(2)`
`[:' ` condition of tangencny of line `y= mx + c to ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`is = a^(2)m^(2)+b^(2)`
here `m=(1)/(2 beta ), a =1, b=-(1)/(sqrt(2)) and c=( beta)/(2)]`
`implies ( beta^(2))/(4) =(1) / ( 4 beta^(2))+(1)/(sqrt(2))`
`implies beta^(4)= 1+ 2 beta^(2)` -1=0`
`implies beta^(2)=(2+- sqrt(4+4))/(2) =(2+- 2sqrt(2))/( 2) = 1 +- sqrt(2) `
` implies beta ^(2)=1+ sqrt(2)=1+ sqrt(2)`