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Find the equation of the ellipse whose i...

Find the equation of the ellipse whose is (5,6), equation of directrix x+y+2=0 and eccentricity is `(1)/(2)`.

Text Solution

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Here the focus ellipse is S(5,6)
Let P(x,y) be a variable point on ellipse and PM is the perpendicular from P to the directrix.
From the definition of ellipse
`PS=e*PM`
`rArr" "sqrt((x-5)^(2)+(y-6)^(2))=(1)/(2)*(x+y+2)/(sqrt(1+1))`
`rArr" "8(x^(2)+y^(2)-10x-12y+61)`
`=(x+y+2)^(2)`
`=x^(2)+y^(2)+4+2xy+4x+4y`
`rArr" "7x^(2)+7y^(2)-2xy-84x-100y+484=0`
which is the required equation of ellipse.
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