Find the equation of the ellipse whose focus is `(1,0),` the directrix is `x+y+1=0` and eccentricity is equal to `1sqrt(2.)`

Options

A. `(x-1)^(2)+y^(2)=(x+y+1)^(2)`
B. `2|(x-1)^(2)+Y^(2)|=(x+y+1)^(2)`
C. `4{(x-1)^(2)+y^(2)}=(x+y+1)^(2)`
D. none of these

Answer

A

Explanation

Let S(1,0) be the focus and ` Z Z'` be the directrix let P(x,y) be any point on the ellipse and PM be perpendicular from P on directrix . Then by definition , we have .
`SP=ePM, ` where `e=1//sqrt(2)`
`implies SP^(2) =e^(2)PM^(2) `
` implies (x-1)^(2) +(y-0)^(2) =((1)/(sqrt(2)))^(2){(x+y+1)/(sqrt(1+1))}^(2)`
`implies 4{(x-1)^(2)+y^(2)}=(x+y+1)^(2)`