The sets S and E are defined as given below:
`S={(x,y): |x-3|lt1and|y-3|lt1}` and
`E={(x,y):4x^(2)+9y^(2)-32x-54y+109le0}`.
Show that `SsubE`.

Options

A. `Asub B`
B. `Bsub A`
C. `A=B`
D. none of these

Explanation

we have ,
`|x-3|lt1 and |y-3|lt1`
`implies 2ltxlt4and 2ltylt4`
` thus ,A is the set of all points (x,y) lying inside the square formed by the lines `x=2,x=4,y=2 and y=4.`
we have
`4x^(2)+9y^(2) -32x-54y+109le0`
`implies4(x^(2)-8x)+9(y^(2)-6y)+109le0`
`implies 4(x-4)^(2)+9(y-3)^(2)le 36`
`implies ((x-4)^(2))/(3^(2))+((y-3)^(2))/(2^(2))le1`
thus,B is set of all point lting inside the ellipse having its centre at (4,3)and maor and minor axes of length the group of two regions 3 and 2 units.
it can be easily seen by drawing the graphs of two regional that `Asub B.`