Nā nīnau hoʻohālike a me ke kūkākūkā ʻana o nā pōʻai a me nā tangents
ʻO nā pōʻai a me nā tangents ʻelua mau kumuhana i kūkākūkā pinepine ʻia ma ka makemakika, ʻoi aku hoʻi ma ka pae kula kiʻekiʻe. ʻO ka hoʻomaopopo ʻana i ke kumumanaʻo a me ka hoʻopili ʻana o nā tangents i nā pōʻai he mea nui ia no ka hoʻoikaika ʻana i kou ʻike i ke geometry. E hāʻawi kēia ʻatikala i nā pilikia hoʻohālike a me nā kūkākūkā ʻana ma nā pōʻai a me nā tangents e hāʻawi i ka poʻe heluhelu i kahi ʻike hohonu.
Hoʻolauna i ke Kumumanaʻo o nā Pōʻai a me nā Tangents
Pōʻai
ʻO ka pōʻai he hui o nā kiko i loko o kahi mokulele e like ka mamao mai kahi kiko paʻa i kapa ʻia ʻo ke kikowaena o ka pōʻai. Ua ʻike ʻia kēia mamao paʻa ʻo ia ka radius o ka pōʻai. Ma ke ʻano makemakika, hiki ke wehewehe ʻia kahi pōʻai e ka hoohalike:
\[ (x – a)^2 + (y – b)^2 = r^2 \]
kahi ʻo \((a, b)\) nā hoʻonohonoho o ke kikowaena o ka pōʻai a ʻo \(r\) ka radius.
ʻO ke kihi
ʻO kahi tangent i kahi pōʻai he laina e pā ana i ka pōʻai ma kahi kiko hoʻokahi. Ua kapa ʻia kēia kiko ʻo ke kiko o ka tangency. ʻO ke ʻano nui o kahi tangent, ʻo ia kona kū pololei ʻana i ka radius i huki ʻia mai ke kikowaena o ka pōʻai a i ke kiko o ka tangency.
Nā Nīnau Laʻana a me ke Kūkākūkā
Nīnau 1: Ke hoʻoholo nei i ka hoohalike o ka laina tangent
Nīnau:
Hāʻawi ʻia kahi pōʻai me ke kikowaena ma \( (2, 3) \) a me ka radius 5. E hoʻoholo i ka hoohalike o ka laina tangent i ka pōʻai ma ke kiko \( P \) me nā hoʻonohonoho \( (5, 7) \).
Kūkākūkā:
KaʻAnuʻu Hana 1: E hōʻoia i ka aia ʻana o ke kiko \(P \) ma ka pōʻai.
No ka nānā ʻana inā aia ʻo \( P (5, 7) \) ma luna o kahi pōʻai me ke kikowaena \( (2, 3) \) a me ke radius \( 5 \), e pani i nā hoʻonohonoho o \( P \) i loko o ka hoohalike o ka pōʻai:
\[ (x – 2)^2 + (y – 3)^2 = 5^2 \]
\[ (5 – 2)^2 + (7 – 3)^2 = 25 \]
3^2 + 4^2 = 25
\[ 9 + 16 = 25 \]
ʻOiai he ʻoiaʻiʻo ke kaulike, aia ke kiko \( P \) ma ka pōʻai.
KaʻAnuʻu Hana 2: E hoʻoholo i ka gradient o ka radius e hele ana ma o \( (2, 3) \) a me \( (5, 7) \):
\[ m_{radius} = \frac{y_2 – y_1}{x_2 – x_1} = \frac{7 – 3}{5 – 2} = \frac{4}{3} \]
KaʻAnuʻu Hana 3: ʻO ke gradient o ka laina tangent e kū pono ana i ka gradient o ka radius (ʻo ke gradient o ka huahana he -1):
\[ m_{tangent} = -\frac{1}{m_{radius}} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4} \]
KaʻAnuʻu Hana 4: E hoʻoholo i ka hoohalike o ka laina tangent me ka hoʻohana ʻana i ke kiko \( P (5, 7) \):
\[ y – y_1 = m (x – x_1) \]
\[ y – 7 = -\frac{3}{4} (x – 5) \]
E hoʻomaʻalahi:
\[ y – 7 = -\frac{3}{4}x + \frac{15}{4} \]
\[ 4y – 28 = -3x + 15 \]
\[ 3x + 4y – 43 = 0 \]
No laila, ʻo ke kaulike o ka laina tangent penei:
\[ 3x + 4y – 43 = 0 \]
Nīnau 2: Ke hoʻoholo nei i ke kiko o ka Tangency mai ka Line Equation
Nīnau:
Hāʻawi ʻia kahi pōʻai me ka hoohalike \( x^2 + y^2 = 25 \) a me kahi laina \( y = \frac{3}{4}x + 2 \). E hoʻoholo i ke kiko o ka tangency ma waena o ka laina a me ka pōʻai.
Kūkākūkā:
KaʻAnuʻu Hana 1: E hoʻololi i ka hoohalike o ka laina i loko o ka hoohalike o ka pōʻai:
Ka hoohalike o kahi pōʻai:
\[ x^2 + y^2 = 25 \]
E hoʻololi iā \( y = \frac{3}{4}x + 2 \) i loko o ka hoohalike pōʻai:
\[ x^2 + \left(\frac{3}{4}x + 2\right)^2 = 25 \]
\[ x^2 + \left(\frac{9}{16}x^2 + \frac{12}{4}x + 4 \right) = 25 \]
\[ x^2 + \frac{9}{16}x^2 + \frac{6}{2}x + 4 = 25 \]
\[ x^2 + \frac{9}{16}x^2 + 3x + 4 = 25 \]
KaʻAnuʻu Hana 2: E hoʻomaʻalahi i ka hoohalike:
\[ 16x^2 + 9x^2 + 48x + 64 = 400 \]
\[ 25x^2 + 48x + 64 – 400 = 0 \]
\[ 25x^2 + 48x – 336 = 0 \]
KaʻAnuʻu Hana 3: Ke ʻimi nei i nā aʻa me ka hoʻohana ʻana i ke ʻano quadratic:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
\[ a = 25, b = 48, c = -336 \]
\[ x = \frac{-48 \pm \sqrt{48^2 – 4 \cdot 25 \cdot (-336)}}{2 \cdot 25} \]
\[ x = \frac{-48 \pm \sqrt{2304 + 33600}}{50} \]
\[ x = \frac{-48 \pm \sqrt{35904}}{50} \]
\[ x = \frac{-48 \pm 189.501}{50} \]
Ke koho ʻana i kahi \( x \) kūpono ma muli o ke kiko tangency (hoʻokahi wale nō \( x \) e hana i kahi kiko tangency):
\[ x = \frac{141.501}{50} \approx 2.83 \]
\[ x \approx 2.83 \]
KaʻAnuʻu Hana 4: E hoʻololi iā \( x \) i loko o ka hoohalike o ka laina e loaʻa ai \( y \):
\[ y = \frac{3}{4}(2.83) + 2 \]
\[ y \approx 2.12 + 2 \]
\[ y \approx 4.12 \]
No laila, ʻo ke kiko o ka tangency ma waena o ka laina \( y = \frac{3}{4}x + 2 \) a me ka pōʻai \( x^2 + y^2 = 25 \) ʻo \( (2.83, 4.12) \).
Ka hopena
ʻO ke akamai i nā manaʻo o nā pōʻai a me nā tangents e pili ana i ka hoʻomaopopo ʻana i nā kumu o ke geometry a me ka hiki ke hoʻoponopono i nā pilikia me ka hoʻohana ʻana i nā kaulike makemakika. Kōkua nā pilikia e like me nā mea ma luna i nā haumāna e hoʻomaʻamaʻa i ka hoʻopili ʻana i ke kumumanaʻo i nā kūlana paʻa. Me ka hoʻomaʻamaʻa mau, manaʻo ʻia nā haumāna e hoʻomaopopo a hoʻoponopono i nā pilikia me ka maʻalahi.
