I wanted to understand how latitude and longitude of a location change by changing the reference ellipsoid. If the location of centre in 2 different ellipsoids is same, and they differ only in the length of minor/major axis, then I don't see how latitude /longitude of a place will differ if any of those 2 ellipsoid is taken as reference.
If two ellipsoids share the same origin and axes (both use Greenwich and the same North pole), but have different axes lengths, only the geodetic latitude will change between the two. This is true because geodetic latitude is defined as a line perpendicular to the ellipsoid surface.
However, you shouldn't normally think in terms of the ellipsoids, but of the geographic/geodetic coordinate reference system (GeoCRS) AKA geodetic datum. A GeoCRS contains a geodetic datum which is in turn defined partially by an ellipsoid.
Current, modern (GNSS-based) GeoCRS will try to have the ellipsoid centered at the earth's center of mass. NAD83 and its realizations/re-adjustments is not quite there as it turns out and that's part of the reason why it's drifting farther from WGS84 and ITRFxx realizations over time. (it's also due to tectonic plate motion among other reasons)
Older GeoCRS / datums like ED50, NAD27, Pulkovo 1942, etc. were all defined locally. I've always explained this as someone chose an origin point, set the direction of North and then fit an ellipsoid to the set of control points being used. That is, the ellipsoid surface is matched to the earth's surface in a local area, not worldwide. That often means that the ellipsoid's origin is not at the earth's center of mass.
One way to model the differences of an older, local datum and a modern one which doesn't have a surface "origin point". Is to figure out the translation values in XYZ (3D Cartesian) between the two ellipsoid's origins, plus sometimes rotation values because the XYZ axes aren't quite the same, and a scale value.
For modern frames, these values were generally be small, sub-meter at worse, because the two frames are trying to be coincident. For modern/older combinations, the translation values can be hundreds of meters different, a few seconds off in rotation, and several parts-per-million different in scale.
This is for non-Earth data, but check out slide 86 on this SEDRIS presentation.
Disclosure: Esri employee and member of subcommittee that maintains the EPSG registry.
The combination of an origin, latitude and longitude defines a vector in three dimensions. This then intersects with the surface of an ellipsoid (or geoid) to define a location on it's surface. Assume you keep the latitude and longitude constant. Imagine the vector as a line from the origin out into space. Then as you change the ratio of the ellipsoid's minor and major axis' the point at which the vector intersects the ellipsoid's surface will change.
If that doesn't work, approach it from the opposite direction. Imagine a location 'A' as a dot on the surface of an ellipsoid that isn't in line with one of it's axis. Now visualise the line connecting this location to the origin, and it's angles to the axis' (i.e. lat and long). If the ellipsoid changes then your dot 'A' will moved in absolute (x,y,z) space. Still with me? This means the angles defining the line connecting the origin to 'A' have also changed.