The locus of the foot of the normal drawn from any point `P(alpha, beta)` to the family of circles `x^(2)+y^(2)-2gx+c=0`, where g is a parameter, is

Options

A. `((5pi)/(6),(7pi)/(6))`
B. `(0,(pi)/(6))`
C. `((11pi)/(6),2pi)`
D. all of these

Explanation

The equation of the hyperbola is `xy=c^(2)`, where `c=2sintehta`. The equation of any tangent to it is `(x)/(t)+yt=x`. If it is normal to each member of the family of circles, it must pass through the centre of each circle i.e. `(1,1)`.
`:.(1)/(t)+t=2cimpliest^(2)-2ct+1=0`
This will have non-real roots, if
`4c^(2)-4 lt 0`
`impliesc^(2)-1 lt 0`
`implies-1 lt c lt 1`
`implies-(1)/(2) lt sintheta lt (1)/(2)` [ `:'c=2sinetheta`]
`impliestheta in (theta,pi//6)uu(5pi//6,7pi//6)uu(11pi//6,2pi)`