Dear Shruti, I guess you want to know the locus of mid-point of intercept made by tangents between the axes.

We know that the equation of line with (h,k) as mid-point of intercept between the coordinate axis is x/2h + y/2k = 1. …... (1)

Equation of tangent to ellipse x2+2y2 = 2 is y = mx±√2m2+1 ….. (2)

Comparing both these equations, we have (2k)2 = 2.k2/h2 + 1

Hence, 1/4y2 + 1/2x2 = 1 is the required answer.

All the best

Hope this helps