The given ellipse has the center at the point (-1,1).

Its major semi-axis is horizontal and have a length 2 = .

Its minor semi-axis is verical and has a length of 1 = .


Now, when you reflect the given ellipse in the line y = -x+12, then, OBVIOUSLY,
the major semi-axis of the reflected ellipse will be VERTICAL of the length 2, and    <<<---=== Notice the difference with @MathLover1 (!)
the minor semi-axis of the reflected ellipse will be HORIZONTAL of the length 1.      <<<---=== Notice the difference with @MathLover1 (!)


Now let's find the coordinates of the CENTER of the reflected ellipse.

The given line is  y = -x + 12, or x + y = 12.

Hence, the perpendicular to this  line has, OBVIOUSLY, the form -x + y = c, where "c" is a constant.

We will find the constant "c", using the condition that the center of the given ellipse (-1,1) lies at the perpendicular line.

-(-1) + 1 = 2 gives the value of "c" as 2;  therefore, the equation of the perpendicular line is -x + y = 2,  or  y = x+2.

The given line  y = -x+12 intersect the "mirror" line y = x+2 at the point x= 5, y= 7.


The point (5,7) is mid-point between the center to (-1,1) and the new center of the reflected ellipse (x,y) on the other side of the given line.

So, the center of the reflected ellipse is the point (11,13).   <<<---=== Notice the difference with @MathLover1 (!)


Thus the equation for the reflected ellipse is

     +  = 1.               <<<---=== Notice the difference with @MathLover1


And, finally, the plot