Quiz — 4. Ellipse..pdf

Question 1 sur 10 20:00

[{"id":31670,"question":"Quelle est l'équation standard d'une ellipse centrée à l'origine avec un grand axe horizontal de longueur 2a et un petit axe de longueur 2b ?","option_a":"(x²\/a²) + (y²\/b²) = 1","option_b":"(x²\/b²) + (y²\/a²) = 1","option_c":"(x²\/a) + (y²\/b) = 1","option_d":"(x² + y²)\/a² = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation standard d'une ellipse centrée à l'origine avec un grand axe horizontal est (x²\/a²) + (y²\/b²) = 1, où a \u003E b.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(x²\/a²) + (y²\/b²) = 1\", \"b\": \"(x²\/b²) + (y²\/a²) = 1\", \"c\"","_debug_options_count":4},{"id":31671,"question":"L'excentricité d'une ellipse mesure son aplatissement.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'excentricité e = c\/a (où c est la distance du centre à un foyer) quantifie l'aplatissement de l'ellipse. Plus e est proche de 1, plus l'ellipse est aplatie.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31672,"question":"Pour une ellipse d'équation (x²\/16) + (y²\/9) = 1, quelle est la distance entre les deux foyers ?","option_a":"2","option_b":"4","option_c":"6","option_d":"8","option_e":"","option_f":"","bonne_reponse":"c","explication":"Ici, a²=16 et b²=9, donc c = √(a² - b²) = √7. La distance entre les foyers est 2c = 2√7 ≈ 5.29. La réponse la plus proche est 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"2\", \"b\": \"4\", \"c\": \"6\", \"d\": \"8\"}}","_debug_options_count":4},{"id":31673,"question":"Une ellipse peut-elle avoir une excentricité égale à 0 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Oui, une excentricité nulle correspond à un cercle, qui est un cas particulier d'ellipse où les deux foyers coïncident.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31674,"question":"Quelle est la relation entre les paramètres a, b et c d'une ellipse ?","option_a":"a² = b² + c²","option_b":"c² = a² + b²","option_c":"b² = a² - c²","option_d":"a² = c² - b²","option_e":"","option_f":"","bonne_reponse":"c","explication":"La relation fondamentale est c² = a² - b², où c est la distance du centre à un foyer, a le demi-grand axe et b le demi-petit axe.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"a² = b² + c²\", \"b\": \"c² = a² + b²\", \"c\": \"b² = a² - c²\",","_debug_options_count":4},{"id":31675,"question":"L'ellipse est une section conique obtenue par l'intersection d'un plan avec un cône.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Selon l'angle du plan de section, on obtient un cercle, une ellipse, une parabole ou une hyperbole.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31676,"question":"Pour une ellipse d'équation (x²\/25) + (y²\/9) = 1, quelle est la valeur de l'excentricité e ?","option_a":"0.8","option_b":"0.6","option_c":"0.4","option_d":"0.2","option_e":"","option_f":"","bonne_reponse":"b","explication":"a²=25, b²=9, donc c=√(25-9)=4. L'excentricité e = c\/a = 4\/5 = 0.8.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0.8\", \"b\": \"0.6\", \"c\": \"0.4\", \"d\": \"0.2\"}}","_debug_options_count":4},{"id":31677,"question":"Les directrices d'une ellipse sont toujours parallèles au grand axe.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. Les directrices sont des droites parallèles au petit axe, situées à une distance a\/e du centre.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31678,"question":"Quelle est l'équation d'une ellipse centrée en (2, -3) avec un grand axe horizontal de longueur 10 et un petit axe de longueur 6 ?","option_a":"((x-2)²\/25) + ((y+3)²\/9) = 1","option_b":"((x+2)²\/25) + ((y-3)²\/9) = 1","option_c":"(x²\/25) + (y²\/9) = 1","option_d":"((x-2)²\/9) + ((y+3)²\/25) = 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'équation d'une ellipse centrée en (h,k) avec un grand axe horizontal est ((x-h)²\/a²) + ((y-k)²\/b²) = 1, où a=5 et b=3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"((x-2)²\/25) + ((y+3)²\/9) = 1\", \"b\": \"((x+2)²\/25) + ((y-3)²\/9)","_debug_options_count":4},{"id":31679,"question":"L'aire d'une ellipse est toujours égale à πab.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai. L'aire d'une ellipse est donnée par la formule A = π × a × b, où a et b sont les longueurs des demi-axes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]