Equation of directrix is 2x−y−1=0.
Let S be the corresponding focus, then S=(−1,2).
Given e=√3
Let P(x,y) be any point on the required hyperbola.
Let PM be the length of the perpendicular from P to the directrix.
Then,PSPM=e=√3
∴PS2=3PM2
or (x+1)2+(y−2)2=3∣∣∣2x−y−1√5∣∣∣2
or 5[(x+1)2+(y−2)2]=3(4x2+y2+1−4xy+2y−4x)
or 5[x2+1+2x+y2−4y+4]=12x2+3y2+3−12xy+6y−12x
⇒5x2+5+10x+5y2−20y+20−12x2−3y2−3+12xy−6y+12x=0
⇒−7x2+2y2+12xy+22x−26y+22=0
or 7x2−2y2−12xy+2x+26y−22=0