Two circles of radii $R=5$ and $r=3$, externally tangent (so the distance between centres is $R+r=8$). The length of a common external tangent between two circles with centre-distance $d$ and radii $R, r$ is

$

L=\sqrt{d^{2}-(R-r)^{2}}.

$

Here $L = \sqrt{8^{2} - 2^{2}} = \sqrt{64 - 4} = \sqrt{60} = 2\sqrt{15}.$

Encoded answer: $a=2,\ b=15,\ a\cdot b = 30.$

Why this is CAT-level: The standard formula is two distinct objects: external tangent uses $(R-r)$, internal tangent uses $(R+r)$. Many students confuse the two -- especially when the circles are already externally tangent (which makes the internal tangent degenerate). Choosing the correct formula is the test. The encoded answer format ($a\cdot b = 30$) blocks accidental matches with $\sqrt{60}$, $4\sqrt{15}$, etc.

Answer: $a\cdot b = 30.$