The locus of mid points of the portions of the tangents to the ellipse x²/a² +y²/b²=1 included between the axes is the curve?

Tangent at any point to the ellipse (acos ß, bsin ß) is xcosß / a + ysinß / b = 1....It mmets yhe axes in points (a secß,0) and (0,b coseß).....Lets the mid point be (h,k) then h= a secß / 2 and k = b cosecß / 2......Rearranging we get, cosß= a/2h and sinß= b/2k....Eliminating ß by squaring and adding.....which gives the locus as::::::::ans... a²/x² + b²/y² = 4.......