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Question

Find the equation of the hyperbola whose one directrix is 2x+y=1, the corresponding focus is (1,2) and eccentricity is 3

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Solution

Equation of directrix is 2x+y1=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 (x1)2+(y2)2=32x+y152
or 5[(x1)2+(y2)2]=3(4x2+y2+1+4xy2y4x)
or 5[x2+12x+y24y+4]=12x2+3y2+3+12xy6y12x
5x2+510x+5y220y+2012x23y2312xy+6y+12x=0
7x2+2y212xy+2x14y+22=0
or 7x22y2+12xy2x+14y22=0

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