This is an active- and placebo-controlled, single-site, three-part trial of MK-1092 in healthy adult participants and in participants with type 1 diabetes mellitus (T1DM). The primary hypothesis for this study is that at a dose with sufficient safety, the mean maximal glucose infusion rate (GIRmax) after single subcutaneous (SC) administration of MK-1092 in adult participants with T1DM is within an acceptable range. (Part 3)