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Question

The equation of the ellipse with focus (-1,1), directrix x-y+3=0 and eccentricity 1/2 is

A
7x2+2xy+7y2+10x+10y+7=0
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B
7x2+7y2+2xy10y+10x+7=0
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C
7x2+2xy+7y2+10x10y7=0
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D
none of these
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Solution

The correct option is B 7x2+7y2+2xy10y+10x+7=0



Let P(x,y) be any point on the ellipse whose focus and eccentricity are S(-1,1) and e=12, respectively.
Let PM be the perpendicular from P on the directrix.
Then SP=e×PM
SP=12×PM
2SP=PM
4(SP)2=PM2
4[(x+1)2+(y1)2]=(xy+312+(1)2)2
4[(x+1+2x+y2+12y)]=x2+y2+92xy6y+6x2
8x2+8+16x+8y2+816y=x2+y2+92xy6y+6x
7x2+7y2+2xy10y+10x+7=0
This is the required equation of the ellipse.

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