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(A) The equation of the director circle of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $x^2 + y^2 = a^2 - b^2$.Given the director circle is

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(A) The equation of the director circle of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $x^2 + y^2 = a^2 - b^2$.Given the director circle is $x^2 + y^2 = 16$,we have $a^2 - b^2 = 16$  $(i)$.The perpendicular distance from the centre $(0,0)$ to the latus rectum of the hyperbola is given by $\frac{a^2}{c}$ or simply the distance to the line $x = ae$,which is $ae$.Given $ae = \sqrt{34}$,then $c^2 = a^2 + b^2 = 34$  $(ii)$.Adding $(i)$ and $(ii)$: $(a^2 - b^2) + (a^2 + b^2) = 16 + 34$ $\Rightarrow 2a^2 = 50$ $\Rightarrow a^2 = 25$ $\Rightarrow a = 5$.Substituting $a^2 = 25$ into $(ii)$: $25 + b^2 = 34$ $\Rightarrow b^2 = 9$ $\Rightarrow b = 3$.Therefore,$a + b = 5 + 3 = 8$.

Let the $X$-axis be the transverse axis and the $Y$-axis be the conjugate axis of a hyperbola $H$. Let $x^2+y^2=16$ be the director circle of $H$. If the perpendicular distance from the centre of $H$ to its latus rectum is $\sqrt{34}$,then $a+b=$

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