Question: What castle ruin in Scotland featured the ancestors of Alexander Hamilton?

Context paragraphs:
Hamilton family: The Hamiltons of the United States are a family of Scottish origin, whose most prominent member was Founding Father Alexander Hamilton (  1755 –1804).  Their ancestors and relations in Scotland included the Lairds of Kerelaw Castle in Stevenston, North Ayrshire, of the Cambuskeith branch of Clan Hamilton.

Terringzean Castle, East Ayrshire: Terringzean Castle, also Taringzean, pronounced 'Tringan', is a Category B listed castle ruin lying above the River Lugar and the Terringzean Holm in the policies of Dumfries House, Parish of Cumnock, Scotland.  The name Craufordstone or Craufurdstoun, has also been used, echoing that it and these lands originally belonged to the Crawfords, as did Leifnorris or Lochnorris Castle which stood on the site of what is now Dumfries House.

Alexander Hamilton (Trumbull): Alexander Hamilton is a 1792 full-length portrait of Alexander Hamilton by John Trumbull.  It is one of multiple paintings John Trumbull made of Alexander Hamilton.

Kerelaw Castle: Kerelaw Castle is a castle ruin situated on the coast of North Ayrshire, Scotland in the town of Stevenston.

Ortnek Castle: Ortnek Castle (Slovene: "Ortneški grad" ), also known as the Old Castle ("Stari grad" ), is a 12th-century castle ruin in the vicinity of the town of Ribnica, Slovenia.  The ruin stands on Big Žrnovec Hill ("Veliki Žrnovec" ), near the village of Hudi Konec.

Meydancık Castle: Meydancık Castle (Turkish: "Meydancık Kale" ) is the name of a castle ruin in Mersin Province, Turkey.  The original name was Kirshu and the name of the ruin during Ottoman times was Beydili Kale.

Schenkenberg Castle: Schenkenberg Castle (German: "Ruine Schenkenberg" ) is a castle ruin above the municipality of Thalheim in the canton of Aargau in Switzerland.  It was built in the 13th Century by the Habsburg dynasty, was the administrative seat for 260 years of a bailiwick of the city of Bern and fell into ruin in the 18th century.

Lihtenberk Castle: Lihtenberk Castle (Slovene: "Grad Lihtenberk" , German: "Schloss Lichtenberg" ) is a 13th-century castle ruin located in the Municipality of Šmartno pri Litiji in central Slovenia, directly adjacent to the later Bogenšperk Castle.  It is best known for a minor association with the 17th-century historian Johann Weikhard von Valvasor, who owned the ruin and styled himself (among other things) "von Lichtenberg."

Kalø Vintage Car Rally: The Kalø Vintage Car Rally is a weekly gathering of vintage car enthusiasts at Kalø Castle Ruin, in Denmark.  Every non-winter Tuesday since 2002, up to 600 vintage cars travel to Kalø, according to founder Bent Ammitsbøl.  The Rally is free for owners and participants.  Events are self-regulated, with no staff or officials.  The rallies are held on a grassy slope bordering the sea at the Bight of Aarhus overlooking Kalø Castle Ruin 2 km from the town of Rønde and 30 km from Denmark's second largest town, Aarhus.  The primary focus is on vintage cars, but old motorcycles and mopeds also participate as do other styles of cars.

Hagaparken: Hagaparken ("Haga Park"), or simply Haga in Solna Municipality just north of Stockholm, Sweden is a vast and popular nature area, with large parks, lakes, woods and gardens.  Within the park is Haga Palace, King Gustav III's Pavilion, the Chinese Pavilion, the Echo Temple, the Turkish Kiosk, an older castle ruin (which is not really a ruin as it is the remains of a castle never finished) and several other interesting buildings on the grounds (such as the peculiar Copper Tents and also the Butterfly House).  Included in the Haga Park is also the Royal Burial Ground of the Swedish Royal family (since 1922), where several members and ancestors of the present Swedish royal Bernadotte family rest.

Supporting paragraphs: Hamilton family Kerelaw Castle
Answer: Kerelaw Castle