A Study of Ramucirumab (LY3009806) or Merestinib (LY2801653) in Advanced or Metastatic Biliary Tract Cancer

Inclusion Criteria: * Have an Eastern Cooperative Oncology Group performance status of 0 or 1. * Have a histologically or cytologically confirmed diagnosis of non-resectable, recurrent, or metastatic biliary tract adenocarcinoma (intrahepatic or extrahepatic cholangiocarcinoma, gallbladder cancer, or Ampulla of Vater) . * Have measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST). * Have adequate biliary drainage. * Have adequate organ function. * Males and females are sterile, postmenopausal, or compliant with a highly effective contraceptive method. * Female participants of childbearing potential must have a negative serum pregnancy test within 7 days prior to first dose. * Are willing to provide blood/serum/plasma and tumor tissue samples for research purposes. Submission of blood/serum/plasma and tumor tissue samples is mandatory for participation in this study, unless restricted per local regulations. Exclusion Criteria: * Previous systemic therapy for locally advanced or metastatic disease is not allowed. * Have a history of or have current hepatic encephalopathy of any grade, or ascites of Grade \>1, or cirrhosis with Child-Pugh Stage B or higher. * Have ongoing or recent (≤6 months) hepatorenal syndrome. * Have had a major surgical procedure or significant traumatic injury including nonhealing wound, peptic ulcer, or bone fracture ≤28 days prior to randomization. * Anticipate having a major surgical procedure during the course of the study. * Has documented brain metastases, leptomeningeal disease, or uncontrolled spinal cord compression. * Within 6 months prior to randomization, have had any arterial thrombotic event, including myocardial infarction, unstable angina, cerebrovascular accident, or transient ischemic attack. * Have an uncontrolled arterial hypertension with systolic blood pressure ≥150 or diastolic blood pressure ≥90 millimeters of mercury (mm Hg) despite standard medical management. * Have a previous malignancy within 5 years of study entry or a concurrent malignancy. * Have a history of gastrointestinal perforation and/or fistulae within 6 months prior to randomization. * Have a known allergy or hypersensitivity reaction to any of the treatment components. * Have a history of uncontrolled hereditary or acquired thrombotic disorder. * Have uncontrolled metabolic disorders or other nonmalignant organ or systemic diseases or secondary effects of cancer that induce a high medical risk and/or make assessment of survival uncertain. * Have mixed hepatocellular biliary tract cancer histology. * Have a corrected QT interval \>470 milliseconds as calculated by the Fridericia equation.