Ke Kūlana o ka Laina e Kūʻē ana i ka Pōʻai
He ʻano geometric kumu ka pōʻai me nā hoʻohana he nui ma nā ʻano like ʻole, mai ka makemakika a i ka physics a me ka ʻenekinia. ʻO kahi ʻano koʻikoʻi o ke aʻo ʻana i nā pōʻai, ʻo ia ka hoʻomaopopo ʻana i ke kūlana o nā laina e pili ana iā lākou. He mea koʻikoʻi kēia kūlana ma nā kūlana he nui, e like me ka hoʻolālā geometric, ka nānā ʻana i ke ʻano, a me ke aʻo ʻana i ka logic a me nā hōʻoia makemakika.
1. Wehewehena o nā Laina a me nā Pōʻai
ʻO ka pōʻai ka hui o nā kiko a pau i loko o kahi mokulele he mamao mau mai kahi kiko waena. ʻO ka laina kahi hui o nā kiko e hana ana i kahi laina pololei palena ʻole.
Ma ke ʻano makemakika, ua hōʻike ʻia kahi pōʻai me ke kikowaena (h, k) a me ka radius r e ka hoohalike:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Hiki ke hōʻike ʻia nā laina ma nā ʻano like ʻole. ʻO ke ʻano maʻamau o kahi laina ma ka mokulele hoʻonohonoho Cartesian penei:
\[ Ax + By + C = 0 \]
2. Ke kūlana o ka laina e pili ana i ka pōʻai
Hiki ke hoʻokaʻawale ʻia ke kūlana o kahi laina e pili ana i kahi pōʻai i ʻekolu mau ʻāpana nui:
1. Laina Pākuʻi
2. Laina Secant
3. Nā Laina ma waho o ka Pōʻai
Laina Pākuʻi i ka Pōʻai
ʻO kahi laina tangent i kahi pōʻai he laina e hui pū ana me ka pōʻai ma hoʻokahi wale nō kiko. Ua kapa ʻia kēia kiko tangency ʻo ke kiko tangency. Ma ke ʻano geometric, he tangent kahi laina i kahi pōʻai inā a inā wale nō:
\[ d = r \]
ʻO d ka mamao mai ke kikowaena o ka pōʻai a i ka laina, a ʻo r ka radius o ka pōʻai.
No ka hoʻoholo ʻana i ka mamao mai ke kikowaena o ka pōʻai (h, k) a i ka laina Ax + By + C = 0, hoʻohana mākou i ke ʻano hana:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Inā \( d = r \), a laila pili ka laina i ka pōʻai.
Kaapuni e hui pū ana me ka laina
Hoʻopili kahi laina i kahi pōʻai ma nā wahi ʻokoʻa ʻelua. I kēia hihia, he secant o ka pōʻai ka laina. Ma ke ʻano makemakika, hoʻopili kahi laina i kahi pōʻai inā ʻoi aku ka mamao mai ke kikowaena o ka pōʻai a i ka laina ma mua o ka radius o ka pōʻai:
\[ d < r \] Ka Laina Ma Waho o ka Pōʻai Aia kahi laina ma waho o ka pōʻai inā ʻoi aku ka mamao mai ke kikowaena o ka pōʻai a i ka laina ma mua o ka radius o ka pōʻai: \[ d > r \]
3. Ka Nānā ʻana i ke Kūlana o kahi Laina e Pili Ana i kahi Pōʻai me nā Laʻana
Eia kekahi laʻana e wehewehe i ka hoʻomaopopo ʻana i ke kūlana o kahi laina e pili ana i kahi pōʻai.
Laʻana 1: Ka Laina Paʻa i kahi Pōʻai
Manaʻo mākou he pōʻai ko mākou me ke kikowaena (3, 2) a me ka radius 5. ʻO ka nīnau inā paha e pili ana ka laina \( x + 2y = 7 \) i ka pōʻai?
ʻO ka hana mua, ʻike mākou i ka mamao mai ke kikowaena o ka pōʻai a i ka laina.
\[ h = 3, \, k = 2, \, r = 5 \]
\[ d = \frac{|1\cdot3 + 2\cdot2 – 7|}{\sqrt{1^2 + 2^2}} = \frac{|3 + 4 – 7|}{\sqrt{5}} = \frac{0}{\sqrt{5}} = 0 \]
ʻOiai ʻo \( d \neq r \), ʻaʻole e pā ka laina i ka pōʻai. E nānā hou kākou ma ka helu ʻana:
\[
d = \frac{|1\cdot3 + 2\cdot2 – 7|}{\sqrt{1^2 + 2^2}} = \frac{|3 + 4 – 7|}{\sqrt{5}} = \frac{0}{\sqrt{5}} = 0
\]
ʻO ka mea pōʻino, no ka laʻana, inā he hewa inā \( d \neq r \), e hoʻāʻo mākou i ka laina \( x + 2y = 8 \)
ʻO ka hana mua, ʻike mākou i ka mamao mai ke kikowaena o ka pōʻai a i ka laina.
\[ h = 3, \, k = 2, \, r = 5 \]
\[ d = \frac{|1\cdot3 + 2\cdot2 – 8|}{\sqrt{1^2 + 2^2}} = \frac{|3 + 4 – 8|}{\sqrt{5}} = \frac{1}{\sqrt{5}} = 4,7 \] a i ʻole d
ʻOiai ʻo \( d < r \), ʻaʻole e hoʻopā ka laina i ka pōʻai. Laʻana 2: Ke Kuʻi nei kahi Laina i kahi Pōʻai I kēia manawa, loaʻa iā mākou kahi pōʻai me ke kikowaena (0, 0) a me ka radius 3. E nānā kākou inā e kuʻi ana ka laina y = x + 1 i ka pōʻai. ʻO ka mea mua, e ʻike kākou i ka mamao mai ke kikowaena o ka pōʻai a i ka laina. \[ h = 0, \, k = 0, \, r = 3 \] \[ A = 1, \, B = -1, \, C = -1 \] \[ d = \frac{|1\cdot0 - 1\cdot0 - 1|}{\sqrt{1^2 + (-1)^2}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \approx 0.71 \] ʻOiai ʻo \( d < r \), e kuʻi ana ka laina i ka pōʻai ma nā wahi ʻelua. 4. Hopena ʻO ke kūlana o kahi laina e pili ana i kahi pōʻai he manaʻo nui ia i ke geometry e pono ai i nā ʻano noi like ʻole. Hiki ke hoʻokaʻawale ʻia kēia kūlana ma muli o ka mamao mai ke kikowaena o ka pōʻai a i ka laina. Inā like ka mamao me ka radius o ka pōʻai, a laila ua tangent ka laina i ka pōʻai. Inā emi ka mamao ma mua o ka radius o ka pōʻai, a laila ua hui ka laina me ka pōʻai. Inā ʻoi aku ka mamao ma mua o ka radius o ka pōʻai, a laila aia ka laina ma waho o ka pōʻai. ʻO ka hoʻomaopopo ʻana i ke kūlana o kahi laina e pili ana i kahi pōʻai e kōkua i kahi ākea o ka nānā ʻana i ke geometric a me nā noi hana ʻē aʻe, mai ka hoʻolālā ʻenekinia a me ka hoʻolālā a hiki i ka noiʻi ʻepekema paʻakikī. Me ka hoʻomaopopo paʻa ʻana i kēia kūlana, hiki i kahi mea hana a mea noiʻi paha ke hoʻolālā a loiloi i nā hale me ka pololei a me ka maikaʻi.
