1. A and B take turns shooting at a target, with A starting first. Let the hit rates of the two people be $a, b \in (0,1)$. It is known that the probability of both hitting the target for the first time is the same, and once A hits the target, he stops shooting. Then the equation that the real numbers $a, b$ need to satisfy is
$-1 \cdot \frac{a}{1-a}=b$.
Since both miss on their first attempt, they are back to the starting point. Therefore, we only need to consider the relationship between the probability of A hitting the target on the first attempt, $a$, and the probability of A missing on the first attempt and B hitting on the first attempt, $(1-a) b$.
According to the problem, the equation $\frac{a}{1-a}=b$ must be satisfied.