We know that for any given conicoids if we translate (shift) the origin (axes) suitably, to new origin (Axes) ,then the coefficients of the linear terms in the conicoids vanishes and here we will use this idea to reduce the coinicoids (i) and (ii).
i). We have given that the non standard form of the conicoid is
which looks like,
Let us consider such a suitable transformation of axes is,
Thus,Old origin and new origin .Now on evaluating the equation and equating the coefficients of the linear terms we get ,Hence the equation reduced to
Thus, standard form of the conicoid will be
Which is a equation of paraboloid.
ii). Given conicoid is
Which looks like,
Now, exactly similar to the question (i) if we transform the axes,
Thus, the standard form is
Which is also a paraboloid.
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