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Find the center, foci, the length of the...

Find the center, foci, the length of the axes, and the eccentricity of the ellipse `2x^2+3y^2-4x-12 y+13=0`

Text Solution

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The given equation uis
`2(x^(2)-2x) +3(y^(2)-4y) +13=0`
`rArr 2(x-1)^(2) +3y-2)^(2)=1`
`rArr((x-1)^(2))/(1//2)+((y-2)^(2))/(1//3)`
Centre of the ellipse is (1/2).
`a^(2)=1//2 and b^(2)=1//3`
Since `agtb`, equation of maajor axis is y=2 and eqution of minor axis is =x 1.
Legth of major axis = `2a=sqrt(2)`
and Length of minor axis `=2b=(2)/(sqrt(3))`
and `e=sqrt(1-(b^(2))/(a^(2)))=sqrt(1-(2)/(3))=(1)/(sqrt(3))`
`a^(2)e^(2)=a^(2)-b^(2)=(1)/(6)`
`:. ae=(1)/(sqrt(6))`
Then foci are `(1+-(1)/(sqrt(6)),2)`
Eqution of directrix are `x-1=+-(a)/(\e)+-(sqrt(3))/(sqrt(2))`
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