Let P(x, y) be a variable point such that
`|sqrt((x-1)^(2)+(y-2)^(2))-sqrt((x-5)^(2)+(y-5)^(2))|=3`
which represents a hyperbola.
IF the origin is shifted to the point (3, 7/2) and the axes are rotated through an angle `theta` in clockwise sense so that the equation of the given hyperbola changes to the standard form `(X^(2))/(a^(2))-(y^(2))/(b^(2))=1`, then `theta` is

Options

A. `(x-3)^(2)+(y-(7)/(2))^(2)=(55)/(4)`
B. `(x-3)^(2)+(y-(7)/(2))^(2)=(25)/(4)`
C. `(x-3)^(2)+(y-(7)/(2))^(2)=(7)/(4)`
D. none of these

Answer

D

Explanation

Director circle is the given by
`(X-h)^(2)+(y-k)^(2)=a^(2)-b^(2)`
the where (h, k) is centre, i.e., the midpoint of foci
`((1+5)/(2),(2+5)/(2))-=(3,(7)/(2))`
`b^(2)=a^(2)(e^(2)-1)=((3)/(2))^(2){((5)/(3))^(2)-1}=4`
Therefore, the firector circle is
`(x-3)^(2)+(y-(7)/(2))^(2)=(9)/(4)-4`
`"or "(x-3)^(2)+((y-7)/(2))^(2)=-(7)/(4)`
This does not represent any real point.