元気 (well/energetic) is a な-adjective; 病気 (illness) is a noun. This changes how each connects to other words.
[Person] + は + 元気(です/な+Noun) | 病気(です/の+Noun)
お元気ですか?
Ogenki desu ka?
Are you doing well?
Standard greeting; the お is a politeness prefix on 元気.
病気で学校を休んだ。
Byouki de gakkou o yasunda.
I took off school due to illness.
病気 (noun) + で (cause/reason) + 学校を休む (skip school).
弟は元気な子どもです。
Otouto wa genki na kodomo desu.
My little brother is an energetic child.
元気 (な-adjective) + な + 子ども — な links it to the noun it describes.
• Catching a cold: You don't 'have' or 'become' a cold — you 引く (pull) it: 風邪を引く. For the current state use 風邪を引いている (I have a cold). • 元気な人 (an energetic person) uses な; 病気の人 (a sick person) uses の because 病気 is a noun.
The JLPT rarely asks whether you recognise a pattern — it asks which of several similar patterns fits. These are the ones 元気/病気 is tested against.
元気 modifies nouns with な (元気な子); 病気 is a noun so it modifies with の (病気の犬).
元気 = lively / in good spirits right now; 健康 = medically healthy (long-term physical health), often used as a noun/な-adjective in formal contexts.
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