Question

What kind of conic is defined by the equation `2x^2+4y^2-4x+12y=0`?

Answer 1

the conic is an ellipse with centre `(1,-3/2)` and eccentricity `1/sqrt2`

Explanation:

the given equation is `2x^2+4y^2-4x+12y=0`
we are basically trying to convert the equation in form of perfect squares.
on doing so the equation can be written as `2(x^2-2x+1)+4(y^2+3y+(3/2)^2)=2+9`
`rArr2(x-1)^2+4(y+3/2)^2=11`
`rArr(2(x-1)^2)/11+(4(y+3/2)^2)/11=1`
`rArr(x-1)^2/(11/2)+(y+3/2)^2/(11/4)=1`
this is of the form `((x-h)^2)/a^2+((y-k)^2)/b^2=1`
which is the general form of an elipse
hence centre is `(h,k)=(1,-3/2)` and eccentricity `e=sqrt((a^2-b^2)/a^2)=1/sqrt2`