(d) $\frac{{ab – {y^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$or ${y^2}\left( {\frac{{{a^2} – {b^2}}}{{{a^2}{b^2}}}} \right) = \frac{{a – b}}{a}$

or ${y^2} = \left( {\frac{{a{b^2}}}{{a + b}}} \right)$and ${x^2} = \left( {\frac{{{a^2}b}}{{a + b}}} \right)\,\, $

$\Rightarrow \,(x,\,y) = \left( {a\sqrt {\frac{b}{{a + b}}} ,b\sqrt {\frac{a}{{a + b}}} } \right)$

Slope of tangent at ellipse $ = \frac{{ – {b^2}x}}{{{a^2}y}} = \frac{{ – {b^2}}}{{{a^2}}}\sqrt {\frac{a}{b}} $

Slope of tangent at circle $ = – \frac{x}{y} = – \sqrt {\frac{a}{b}} $

$\therefore \theta = {\tan ^{ – 1}}\left[ {\frac{{ – \sqrt {\frac{a}{b}} + \frac{{{b^2}}}{{{a^2}}}\sqrt {\frac{a}{b}} }}{{1 + \frac{{{b^2}}}{{{a^2}}}.\frac{a}{b}}}} \right]$

$i.e.$, $\theta = {\tan ^{ – 1}}\left[ {\frac{{a – b}}{{\sqrt {ab} }}} \right]$.