$\frac{ x ^{2}}{ a ^{2}}+\frac{ y ^{2}}{ b ^{2}}=1, x ^{2}+ y ^{2}= ab$
$\frac{2 x _{1}}{ a ^{2}}+\frac{2 y _{1} y ^{\prime}}{ b ^{2}}=0$
$\Rightarrow y _{1}^{\prime}=\frac{- x _{1}}{ a ^{2}} \frac{ b ^{2}}{ y _{1}}...$(1)
$\therefore 2 x _{1}+2 y _{1} y ^{\prime}=0$
$\Rightarrow y _{2}^{\prime}=\frac{- x _{1}}{ y _{1}}...$(2)
Here $\left(x_{1} y_{1}\right)$ is point of intersection of both curves
$\therefore x _{1}^{2}=\frac{ a ^{2} b }{ a + b }, y _{1}^{2}=\frac{ ab ^{2}}{ a + b } $
$\therefore \tan \theta=\left|\frac{ y _{1}^{\prime}- y _{2}^{\prime}}{1+ y _{1}^{\prime} y _{2}^{\prime}}\right|=\left|\frac{\frac{- x _{1} b ^{2}}{ a ^{2} y _{1}}+\frac{ x _{1}}{ y _{1}}}{1+\frac{ x _{1}^{2} b ^{2}}{ a ^{2} y _{1}^{2}}}\right|$
$\tan \theta=\left|\frac{-b^{2} x_{1} y_{1}+a^{2} x_{1} y_{1}}{a^{2} y_{1}^{2}+b^{2} x_{1}^{2}}\right|$
$\tan \theta=\left|\frac{a-b}{\sqrt{a b}}\right|$